{"id":86,"date":"2021-12-15T09:53:22","date_gmt":"2021-12-15T09:53:22","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/die-ableitung\/"},"modified":"2021-12-15T09:53:22","modified_gmt":"2021-12-15T09:53:22","slug":"die-ableitung","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/die-ableitung\/","title":{"raw":"Die Ableitung","rendered":"Die Ableitung"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z4c1bcff6261e\" class=\"sectionHead\"><span class=\"titlemark\">8.1 <\/span> <a id=\"x1-2260001\"><\/a>Die Ableitung<\/h3> <a id=\"x1-226001r224\"><\/a> <h4 id=\"z1323e52baf31\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.1 <\/span> <a id=\"x1-2270001\"><\/a>Definition und geometrische Interpretation<\/h4> <p class=\"noindent\">Eine (nicht-vertikale) <span class=\"ecbx-1095\">Gerade <\/span>im <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> ist eine Teilmenge der Form <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>m<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> f\u00fcr Parameter <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> oder alternativ ausgedr\u00fcckt der Graph der (affinen) Abbildung <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mi>m<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Meist nennt man Funktionen dieser Form ebenfalls Geraden. Der Parameter <math display=\"inline\"><mi>m<\/mi><\/math> der Geraden <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>m<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><\/math> wird auch die <span class=\"ecbx-1095\">Steigung <\/span>der Geraden genannt. Wir m\u00f6chten uns nun mit Funktionen besch\u00e4ftigen, die sich um einen Punkt im Definitionsbereich durch Geraden approximieren lassen. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z1d75e9ac9cb9\"> <a id=\"x1-227001r1\"><\/a> <span class=\"ecbx-1095\">Definition 8.1 <\/span>(Differenzierbarkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge, <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine Funktion und <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein H\u00e4ufungspunkt von <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecbx-1095\">differenzierbar <\/span>ist, falls der Grenzwert                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-227002r1\" \/><mstyle class=\"maketag\"><mtext>(8.1)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">existiert. In diesem Fall nennen wir <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die <span class=\"ecbx-1095\">Ableitung <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> bei <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>f<\/mi><\/math> bei jedem H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> in <math display=\"inline\"><mi>D<\/mi><\/math> differenzierbar ist, dann sagen wir auch, dass <math display=\"inline\"><mi>f<\/mi><\/math> (auf <math display=\"inline\"><mi>D<\/mi><\/math>) <span class=\"ecbx-1095\">differenzierbar <\/span>ist und nennen die Funktion <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> definiert auf den H\u00e4ufungspunkten von&nbsp;<math display=\"inline\"><mi>D<\/mi><\/math> in&nbsp;<math display=\"inline\"><mi>D<\/mi><\/math> die <span class=\"ecbx-1095\">Ableitung <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Falls <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein rechtseitiger H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> ist, dann ist <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecbx-1095\">rechtsseitig differenzierbar<\/span>, wenn die <span class=\"ecbx-1095\">rechtsseitige Ableitung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">existiert. <span class=\"ecbx-1095\">Linksseitige Differenzierbarkeit <\/span>und die <span class=\"ecbx-1095\">linksseitige<\/span> <span class=\"ecbx-1095\">Ableitung<\/span>&nbsp;<math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> werden analog \u00fcber die Bewegung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2197<\/mo> <mi>a<\/mi><\/math> definiert.                                                                                                                                                                           <\/p> <\/div> <p class=\"indent\">Wir nennen <math display=\"inline\"><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>h<\/mi><\/math> im Zusammenhang mit der Definition in (<a href=\"..\/..\/chapter\/die-ableitung#x1-227002r1\">8.1<\/a>) auch das <span class=\"ecbx-1095\">Inkrement des Arguments <\/span>oder der <span class=\"ecbx-1095\">unabh<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">ngigen<\/span> <span class=\"ecbx-1095\">Variablen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das <span class=\"ecbx-1095\">Inkrement der Funktion<\/span> und <math display=\"inline\"><mfrac><mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>f<\/mi><\/mrow> <mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> den <span class=\"ecbx-1095\">Differenzenquotienten<\/span>. Die Ableitung von <math display=\"inline\"><mi>f<\/mi><\/math> bei <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">,<\/span> welche in dieser Formulierung der Grenzwert des Differenzenquotienten <math display=\"inline\"><mfrac><mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>f<\/mi><\/mrow> <mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> f\u00fcr <math display=\"inline\"><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>0<\/mn><\/math> ist, schreibt man auch als <math display=\"inline\"><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und nennt dies den <span class=\"ecbx-1095\">Differentialquotienten <\/span>(in der Leibniz-Notation). Weiters nennt man <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> auch die <span class=\"ecbx-1095\">Ableitung<\/span> <span class=\"ecbx-1095\">nach <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> was vor allem dann n\u00fctzlich ist, wenn <math display=\"inline\"><mi>f<\/mi><\/math> auch von weiteren Parametern abh\u00e4ngen darf. <\/p><p class=\"indent\">Wir m\u00f6chten aber betonen, dass <math display=\"inline\"><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> nicht als Quotient, sondern nur als Grenzwert von Quotienten definiert wurde. Falls die unabh\u00e4ngige Variable <math display=\"inline\"><mi>t<\/mi><\/math> (f\u00fcr Zeit) und nicht <math display=\"inline\"><mi>x<\/mi><\/math> ist, dann verwendet man manchmal auch die Notation <math display=\"inline\"><mi>\u1e8b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u1e8f<\/mi> <\/math> f\u00fcr die Ableitung von Funktionen <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Eine weitere Schreibweise der Definition in (<a href=\"..\/..\/chapter\/die-ableitung#x1-227002r1\">8.1<\/a>) ist in der Landau-Notation (siehe Abschnitt <a href=\"..\/..\/chapter\/landau-notation#x1-1820006\">6.6<\/a>) <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>a<\/mi><\/math> oder \u00e4quivalenterweise <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-227003r2\" \/><mstyle class=\"maketag\"><mtext>(8.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Hierbei wird die Funktion <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das <span class=\"ecbx-1095\">Differential <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> genannt und die Gerade <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die <span class=\"ecbx-1095\">affine <\/span>oder <span class=\"ecbx-1095\">lineare<\/span> <span class=\"ecbx-1095\">Approximation <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> oder die <span class=\"ecbx-1095\">Tangente <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> bei <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">,<\/span> siehe auch Figur&nbsp;<a href=\"..\/..\/chapter\/die-ableitung#x1-227005r1\">8.1<\/a>. Wir erinnern daran, dass wir in (<a href=\"..\/..\/chapter\/die-ableitung#x1-227003r2\">8.2<\/a>) <math display=\"inline\"><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> als Platzhalter einer Funktion (welcher?) interpretieren, die f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>a<\/mi><\/math> schneller abf\u00e4llt als <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist wegen (<a href=\"..\/..\/chapter\/die-ableitung#x1-227003r2\">8.2<\/a>) <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <math display=\"inline\"><mi>f<\/mi><\/math> ist bei <math display=\"inline\"><mi>a<\/mi><\/math> stetig, wenn <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> differenzierbar ist. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze631e7dd7186\"> <a id=\"x1-227004r2\"><\/a> <span class=\"ecbx-1095\">Applet 8.2 <\/span>(Bewegung der Sekante)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"550px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/N6Q4PScN\/width\/688\/height\/550\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir sehen den Graphen einer Funktion und wie die Sekante zwischen <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><\/math> <span class=\"ecti-1095\">sich bei den meisten Fusspunkten <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">der Tangente bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hert falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>h<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=565&amp;url=Pictures\/ableitung\/def.pdf\"><img id=\"zfbbe15b946f3\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/ableitung\/def.svg\" width=\"565\"><\/div> <a id=\"x1-227005r1\"><\/a> <a id=\"x1-227006\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;8.1:              <\/span><span class=\"content\">Die              geometrische              Interpretation               der          Ableitung          einer          reellwertigen          Funktion               <math display=\"inline\"><mi>f<\/mi><\/math>             bei&nbsp;<math display=\"inline\"><mi>a<\/mi><\/math>             ist     die     Steigung     der     Tangenten     des     Graphen     bei               <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span>               Denn wenn <math display=\"inline\"><mi>x<\/mi><\/math>             gegen <math display=\"inline\"><mi>a<\/mi><\/math>             strebt,           wird           die           Sekante,           die           durch               <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math>               und <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math>               geht       und       den       Differenzenquotienten       als       Steigung               besitzt,     immer     mehr     zur     Tangente     des     Graphen     bei               <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\">H\u00e4ufig wird in diesem Kapitel (und dem n\u00e4chsten) der Definitionsbereich <math display=\"inline\"><mi>D<\/mi><\/math> der betrachteten                                                                                                                                                                           Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ein Intervall <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>I<\/mi><\/math> mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> sein. Dies hat den Vorteil, dass jeder Punkt in <math display=\"inline\"><mi>I<\/mi><\/math> ein H\u00e4ufungspunkt ist (wieso?) und es somit f\u00fcr jeden Punkt in <math display=\"inline\"><mi>I<\/mi><\/math> Sinn macht, nach der Differenzierbarkeit von <math display=\"inline\"><mi>f<\/mi><\/math> bei diesem Punkt zu fragen. Wir wollen dies aber weder in der Definition noch in den zu besprechenden Ableitungsregeln voraussetzen, damit wir beispielsweise auch von der Ableitung der Funktion <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> sprechen k\u00f6nnen. <\/p><p class=\"indent\">Meist werden wir reellwertige Funktionen betrachten. Doch wird es teilweise n\u00fctzlich sein, den Begriff der Ableitung und manche der Gesetze auch f\u00fcr komplexwertige Funktionen verwenden zu k\u00f6nnen. Wir bemerken also, dass Definition <a href=\"..\/..\/chapter\/die-ableitung#x1-227001r1\">8.1<\/a> analog auch f\u00fcr komplexwertige Funktionen verwendet werden kann. Wie in Abschnitt <a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-1480004\">5.3.4<\/a> l\u00e4uft dies darauf hinaus, dass sowohl Real- als auch Imagin\u00e4rteil differenzierbar sein sollten. <\/p><p class=\"indent\">Schlussendlich wollen wir noch anmerken, dass die Ableitung eine rein lokale Operation darstellt. Genauer gesagt, angenommen <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ist ein H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> sind bei&nbsp;<math display=\"inline\"><mi>a<\/mi><\/math> differenzierbare Funktionen, so dass es ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt mit <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dies ergibt sich unmittelbar aus der Definition der Grenzwerte, die <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> definieren (wieso?). Wir werden dies im Folgenden teils implizit verwenden. <a id=\"x1-227007r227\"><\/a> <\/p> <h4 id=\"za345598f62ae\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.2 <\/span> <a id=\"x1-2280002\"><\/a>Beispiele und Ableitungsregeln<\/h4> <p class=\"noindent\">Wir wollen nun zeigen, dass viele der uns gel\u00e4ufigen Funktionen differenzierbar sind und dass wir die Ableitung (meistens) mittels einigen konkreten Gesetzen bestimmen k\u00f6nnen. Wir beginnen aber zuerst mit elementaren Beispielen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb2f92347a7d0\"> <a id=\"x1-228001r3\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.3 <\/span>(Erste Beispiele differenzierbarer Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Konstante  Funktionen  sind  <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berall  differenzierbar  und  haben  die  Nullfunktion  als<\/span> <span class=\"ecti-1095\">Ableitung (wieso?).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tsfunktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist differenzierbar und ihre Ableitung ist die konstante<\/span> <math display=\"inline\"><mn>1<\/mn><\/math><span class=\"ecti-1095\">-Funktion,<\/span> <span class=\"ecti-1095\">denn<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Exponentialfunktion <\/span><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist differenzierbar und ihre Ableitung ist die Exponentialfunktion. Allgemeiner behaupten wir, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">ein festes<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">(oder<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><span class=\"ecti-1095\">) die<\/span> <span class=\"ecti-1095\">Ableitung von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gegeben<\/span><span class=\"ecti-1095\">&nbsp;ist. In<\/span> <span class=\"ecti-1095\">der Tat gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>h<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><msup><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>h<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">da die Abbildung <\/span><math display=\"inline\"><mi>h<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup> <mfrac><mrow><msup><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>h<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">nach Satz <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\"><span class=\"ecti-1095\">7.56<\/span><\/a> <span class=\"ecti-1095\">stetig ist.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">Wir besprechen weitere Beispiele von differenzierbaren Funktionen und ein Beispiel einer nicht-differenzierbaren Funktion in der folgenden \u00dcbung. <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z4f9bd5b67a64\"> <a id=\"x1-228005r4\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.4 <\/span>(Weitere differenzierbare Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie die Additionstheoreme aus Abschnitt <\/span><a href=\"..\/..\/chapter\/trigonometrische-funktionen#x1-2090001\"><span class=\"ecti-1095\">7.6.1<\/span><\/a> <span class=\"ecti-1095\">(oder Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228001r3\"><span class=\"ecti-1095\">8.3<\/span><\/a> <span class=\"ecti-1095\">(iii)), um zu<\/span> <span class=\"ecti-1095\">zeigen, dass der Sinus und der Kosinus differenzierbare Funktionen sind und die<\/span> <span class=\"ecti-1095\">Ableitungsregeln<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gelten.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Funktionen <\/span><math display=\"inline\"><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <\/math> <span class=\"ecti-1095\">differenzierbar sind und verifizieren Sie die Ableitungsregeln<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Betragsfunktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">nicht differenzierbar ist und bestimmen Sie bei jedem Punkt in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die linksseitige und die rechtsseitige Ableitung.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Wie in (ii) und (iii) von \u00dcbung <a href=\"..\/..\/chapter\/die-ableitung#x1-228005r4\">8.4<\/a> schon verwendet, wollen wir f\u00fcr Funktionen wie zum Beispiel die Funktion <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> die durch Formeln gegeben sind, nicht immer einen Namen einf\u00fchren, um die Ableitung hinschreiben zu k\u00f6nnen. Stattdessen schreiben wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und meinen damit, dass die Funktion <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math> auf ihrem maximalen Definitionsbereich differenzierbar ist und dass ihre Ableitung bei <math display=\"inline\"><mi>x<\/mi><\/math> durch <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> gegeben ist. Insbesondere ist&nbsp;<math display=\"inline\"><mi>x<\/mi><\/math> in obiger Gleichung nicht als Zahl, sondern vielmehr als Argument der Funktion und der Ableitung zu erachten. <\/p><p class=\"indent\">Wie schon bei stetigen und Riemann-integrierbaren Funktionen m\u00f6chten wir nicht immer von Hand zeigen m\u00fcssen, dass eine gegebene Funktion differenzierbar ist. Stattdessen wollen wir allgemeine Regeln beweisen, auf die sich die Differenzierbarkeit verschiedener Funktionen zur\u00fcckf\u00fchren l\u00e4sst. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z4bd25f5e8523\"> <a id=\"x1-228010r5\"><\/a> <span class=\"ecbx-1095\">Proposition 8.5 <\/span>(Summen und Produkte differenzierbarer Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Teilmenge und <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bei<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar.<\/span> <span class=\"ecti-1095\">Dann sind <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">bei<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist jedes skalare Vielfache von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dies gilt ebenso f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r komplexwertige Funktionen.<\/span> <\/p> <\/div> <p class=\"indent\">Somit bilden die bei <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> differenzierbaren reellwertigen Funktionen einen Unterraum des Vektorraums <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> der reellwertigen Funktionen von <math display=\"inline\"><mi>D<\/mi><\/math> nach <math display=\"inline\"><mi>\u211d<\/mi><\/math> und die Ableitung bei <math display=\"inline\"><mi>a<\/mi><\/math> ist eine lineare Abbildung von diesem Unterraum nach <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Ableitungsregel f\u00fcr das Produkt zweier Funktionen wird auch die <span class=\"ecbx-1095\">Produktregel<\/span> genannt. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir berechnen unter Verwendung der Eigenschaften des Grenzwerts in Abschnitt&nbsp;<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-1750001\">6.4.1<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da <math display=\"inline\"><mi>g<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> stetig ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z92b4e6b65f0e\"> <a id=\"x1-228011r6\"><\/a> <span class=\"ecbx-1095\">Korollar 8.6 <\/span>(Differenzierbarkeit von Polynomen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Reelle Polynome sind auf ganz <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-228012r3\" \/><mstyle class=\"maketag\"><mtext>(8.3)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Nach Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a> und Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\">8.6<\/a> ist insbesondere die Ableitung eines Polynoms wieder ein Polynom. Weiters ist <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> eine lineare Abbildung. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis von Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\">8.6<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Die F\u00e4lle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> wurden bereits in Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-228001r3\">8.3<\/a> besprochen. Wir beweisen (<a href=\"..\/..\/chapter\/die-ableitung#x1-228012r3\">8.3<\/a>) per Induktion nach <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> Angenommen f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> Dann folgt aus Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a>, dass <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> differenzierbar ist und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">erf\u00fcllt, was den Induktionsbeweis abschliesst. Differenzierbarkeit eines beliebigen Polynoms folgt nun aus der Linearit\u00e4t der Ableitung in Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zd918db443989\"> <a id=\"x1-228013r7\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.7 <\/span>(Potenzregel mittels Binomialsatz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie Korollar <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\"><span class=\"ecti-1095\">8.6<\/span><\/a> <span class=\"ecti-1095\">direkt unter Verwendung des Binomialsatzes. Beweisen Sie des<\/span> <span class=\"ecti-1095\">Weiteren,              dass              der              Kern              der              Abbildung<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">aus den konstanten Polynomen besteht. Sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter werden wir sehen, dass nicht nur Polynome<\/span> <span class=\"ecti-1095\">mit Ableitung      Null,      sondern      auch      differenzierbare      Funktionen      auf<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit Ableitung Null konstant sein m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen.<\/span> <\/p> <\/div> <p class=\"indent\">Wieder in Analogie zur Diskussion von stetigen Funktionen (genauer Proposition <a href=\"..\/..\/chapter\/stetigkeit#x1-94011r52\">3.52<\/a>) wollen wir zeigen, dass die Verkn\u00fcpfung zweier differenzierbaren Funktionen auch differenzierbar&nbsp;ist. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z0ea6b008f69c\"> <a id=\"x1-228014r8\"><\/a> <span class=\"ecbx-1095\">Satz 8.8 <\/span>(Kettenregel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>E<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Teilmengen<\/span> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt. Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>E<\/mi><\/math> <span class=\"ecti-1095\">eine bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbare<\/span> <span class=\"ecti-1095\">Funktion, so dass <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><math display=\"inline\"><mi>E<\/mi><\/math> <span class=\"ecti-1095\">ist, und sei <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>E<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbare<\/span> <span class=\"ecti-1095\">Funktion. Dann ist <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbar und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Wir bemerken, dass man zwar versucht sein mag, f\u00fcr den Beweis der Kettenregel den Differenzenquotienten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> zu erweitern. Dies ist im Allgemeinen aber nicht erlaubt, da wir nicht ausschliessen k\u00f6nnen, dass <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr gewisse Punkte <math display=\"inline\"><mi>x<\/mi><\/math> nahe bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ist. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verwenden stattdessen die Umformulierung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> oder genauer formuliert <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei die Funktion <math display=\"inline\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><\/math> auf <math display=\"inline\"><mi>D<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> gegeben ist und bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> stetig ist. Ebenso gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei die bei <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> stetige Funktion <math display=\"inline\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><\/math> auf <math display=\"inline\"><mi>E<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mfrac><mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>y<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>E<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>E<\/mi><\/math> gegeben ist. Zusammen ergibt sich durch Einsetzen von <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/> <mo class=\"MathClass-bin\">+<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span> womit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><\/mtd> <mtd class=\"align-even\"><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wie gew\u00fcnscht. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Abgesehen von Summen, Produkten und Verkn\u00fcpfungen von differenzierbaren Funktionen, m\u00f6chten wir zeigen, dass Quotienten von differenzierbaren Funktionen differenzierbar sind. Wir beginnen dazu mit einem wichtigen Beispiel. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zfbc6700c3814\"> <a id=\"x1-228015r9\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.9 <\/span>(Kehrwert)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar<\/span> <span class=\"ecti-1095\">und es gilt <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">In der Tat ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wegen der Stetigkeit von <\/span><math display=\"inline\"><mi>h<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z4ca95e2cf962\"> <a id=\"x1-228016r10\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.10 <\/span>(Negative Potenzen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Unter Kombination der Kettenregel und Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-228015r9\">8.9<\/a> erh\u00e4lt man nun folgendes Korollar. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zd90a013a1ed5\"> <a id=\"x1-228017r11\"><\/a> <span class=\"ecbx-1095\">Korollar 8.11 <\/span>(Quotientenregel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge,<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt<\/span> <span class=\"ecti-1095\">und seien <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bei<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar.<\/span> <span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, dann<\/span> <span class=\"ecti-1095\">ist auch <\/span><math display=\"inline\"><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>g<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Man beachte, dass der (nat\u00fcrliche) Definitionsbereich der Funktion <span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> der in obigem Korollar nicht erw\u00e4hnt wurde, die Teilmenge <math display=\"inline\"><mi>E<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> ist. Da <math display=\"inline\"><mi>g<\/mi><\/math> beim Punkt <math display=\"inline\"><mi>a<\/mi><\/math> differenzierbar ist, ist <math display=\"inline\"><mi>g<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> stetig. Insbesondere ist, da <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> ist, <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle&nbsp;<math display=\"inline\"><mi>x<\/mi><\/math> nahe genug bei <math display=\"inline\"><mi>a<\/mi><\/math> und <math display=\"inline\"><mi>a<\/mi><\/math> ist ein H\u00e4ufungspunkt von <span class=\"maperiod\"><math display=\"inline\"><mi>E<\/mi><\/math><\/span><span class=\"period\">.<\/span> Damit macht es auch Sinn, von Differenzierbarkeit von <math display=\"inline\"><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> zu sprechen. <\/p><p class=\"indent\">Eine direkte Konsequenz von Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228017r11\">8.11<\/a> ist, dass rationale Funktionen differenzierbar sind, wo definiert. Wir erinnern daran, dass eine rationale Funktion eine Funktion der Form <math display=\"inline\"><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/math> ist, wobei <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> reelle Polynome sind und <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> nicht das Nullpolynom ist. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis von Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228017r11\">8.11<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Es bezeichne <math display=\"inline\"><mi>\u03c8<\/mi><\/math> die Funktion <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> welche nach Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-228015r9\">8.9<\/a> differenzierbar ist. Wir kombinieren dies mit der Kettenregel (Satz <a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\">8.8<\/a>) und erhalten,                                                                                                                                                                           dass die Funktion <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c8<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>g<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> differenzierbar ist mit Ableitung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Verwenden wir nun die Produktregel in Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a>, so ergibt sich, dass <math display=\"inline\"><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> differenzierbar ist und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>g<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>g<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">erf\u00fcllt, was zu zeigen war. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Die Kettenregel erlaubt uns die Berechnung der Ableitung von beliebig kompliziert anmutenden konkreten Beispielen, wobei man stur von aussen nach innen vorgeht wie in folgendem Beispiel. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z91d4528b7bed\"> <a id=\"x1-228018r12\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.12 <\/span>(Vierfach verschachtelte Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir bestimmen die Ableitung der Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">mittels mehrmaligem Anwenden der Kettenregel (Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\"><span class=\"ecti-1095\">8.8<\/span><\/a><span class=\"ecti-1095\">). Da<\/span> <math display=\"inline\"><msup><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <\/math> <span class=\"ecti-1095\">erhalten wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Ebenso<\/span> <span class=\"ecti-1095\">ist wegen <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>h<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>h<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>h<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>h<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mn>2<\/mn><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dadurch erhalten wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mn>2<\/mn><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z697f483756fa\"> <a id=\"x1-228019r13\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.13 <\/span>(Nochmals vierfach verschachtelt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Bestimmen Sie die Ableitung von der Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Unsere vorl\u00e4ufig letzte allgemeine Ableitungsregel betrifft die Ableitung der Umkehrabbildung (siehe dazu auch Satz <a href=\"..\/..\/chapter\/der-satz-ueber-die-umkehrabbildung#x1-97001r64\">3.64<\/a> \u00fcber die Existenz einer stetigen Umkehrabbildung). <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z236d51b630a5\"> <a id=\"x1-228020r14\"><\/a> <span class=\"ecbx-1095\">Satz 8.14 <\/span>(Differenzierbarkeit der inversen Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>E<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Teilmengen und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>E<\/mi><\/math> <span class=\"ecti-1095\">eine stetige, bijektive Abbildung, deren inverse Abbildung<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-punc\">:<\/mo> <mi>E<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ebenfalls stetig<\/span> <span class=\"ecti-1095\">ist. Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in dem<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar ist und <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">gilt, dann ist <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=424&amp;url=Pictures\/ableitung\/invfctder.pdf\"><img id=\"z49d736d8a7f3\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/ableitung\/invfctder.svg\" width=\"424\"><\/div> <a id=\"x1-228021r2\"><\/a> <a id=\"x1-228022\"><\/a> <br><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;8.2:           <\/span><span class=\"content\">Eine           intuitive           Darstellung           von               Satz        <a href=\"..\/..\/chapter\/die-ableitung#x1-228020r14\">8.14<\/a>.        Spiegelt        man        den        Graphen        von               <math display=\"inline\"><mi>f<\/mi><\/math>             und die Tangente beim Punkt <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math>               um die Gerade <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/math>             in <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span>               so erh\u00e4lt man den Graphen von <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math>                   und,      das      ist      die      Behauptung,      die      Tangente      bei               <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span>               Eine          kurze          Rechnung          zeigt,          dass          die               Spiegelung            einer            Gerade            mit            Steigung               <math display=\"inline\"><mi>m<\/mi><\/math>             um <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/math>             Steigung <math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac><\/math>                hat.                                                                                        &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir bemerken zuerst, dass <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> ein H\u00e4ufungspunkt von <math display=\"inline\"><mi>E<\/mi><\/math> ist, womit man von Differenzierbarkeit bei <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> sprechen darf. Tats\u00e4chlich ist nach Annahme <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein H\u00e4ufungspunkt und es existiert eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> mit <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>f<\/mi><\/math> stetig ist, gilt <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> und da <math display=\"inline\"><mi>f<\/mi><\/math> bijektiv ist, gilt <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> eine Folge in <span class=\"maperiod\"><math display=\"inline\"><mi>E<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> die gegen <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> konvergiert. Dann strebt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gegen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> da <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <\/math> per Annahme stetig ist, und es gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach der Charakterisierung der Konvergenz einer Funktion mittels Folgen in Lemma <a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\">6.40<\/a>. Da dies aber f\u00fcr jede Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> wie oben gilt, folgt der Satz wiederum aus Lemma <a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\">6.40<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z481f18f48603\"> <a id=\"x1-228023r15\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.15 <\/span>(Differenzierbarkeit des Logarithmus und der Potenzfunktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist<\/span> <span class=\"ecti-1095\">differenzierbar mit Ableitung <\/span><math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">gegeben durch <\/span><math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><span class=\"ecti-1095\">. Denn die Abbildung<\/span> <math display=\"inline\"><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">:<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist die Umkehrabbildung<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo>  <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und damit folgt<\/span> <span class=\"ecti-1095\">aus Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228020r14\"><span class=\"ecti-1095\">8.14<\/span><\/a><span class=\"ecti-1095\">, dass <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">bei allen Punkten <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">differenzierbar ist mit <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo>  <\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">folgt nun<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist<\/span> <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Also folgt<\/span> <span class=\"ecti-1095\">Differenzierbarkeit von <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>y<\/mi><\/math> <span class=\"ecti-1095\">sowie<\/span> <span class=\"ecti-1095\">die Formel <\/span><math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">aus der Kettenregel (Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\"><span class=\"ecti-1095\">8.8<\/span><\/a><span class=\"ecti-1095\">).<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein beliebiges <\/span><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist die Abbildung <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">In der Tat gilt <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">per Definition beliebiger Potenzen in Abschnitt <\/span><a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-2050002\"><span class=\"ecti-1095\">7.5.2<\/span><\/a><span class=\"ecti-1095\">. Aus Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228001r3\"><span class=\"ecti-1095\">8.3<\/span><\/a> <span class=\"ecti-1095\">und der Ableitung der<\/span> <span class=\"ecti-1095\">Logarithmusabbildung folgt somit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>s<\/mi><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mi>s<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <\/div> <a id=\"x1-228026r228\"><\/a> <h4 id=\"z2b16b3834835\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.3 <\/span> <a id=\"x1-2290003\"><\/a>Extremwerte<\/h4> <p class=\"noindent\">Wie wir in diesem Abschnitt sehen werden, ist die Ableitung auch n\u00fctzlich, um Punkte zu finden, bei denen eine Funktion <math display=\"inline\"><mi>f<\/mi><\/math> ihre Maxima und ihre Minima annimmt. Genauer kann man damit die in folgender Definition eingef\u00fchrten Punkte finden. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z164edf7ef666\"> <a id=\"x1-229001r16\"><\/a> <span class=\"ecbx-1095\">Definition 8.16 <\/span>(Lokale Extremwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ein <span class=\"ecbx-1095\">lokales Maximum <\/span>in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> annimmt, falls es eine Umgebung <math display=\"inline\"><mi>U<\/mi><\/math> von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> in <math display=\"inline\"><mi>D<\/mi><\/math> gibt, auf der <math display=\"inline\"><mi>f<\/mi><\/math> durch <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> beschr\u00e4nkt ist. Genauer formuliert heisst dies, dass es ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt, so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Falls es sogar ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt, so dass <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gilt, dann nimmt <math display=\"inline\"><mi>f<\/mi><\/math> in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein <span class=\"ecbx-1095\">isoliertes lokales Maximum <\/span>an. Der Wert <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math>                                                                                                                                                                           wird auch ein <span class=\"ecbx-1095\">lokales Maximum <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> genannt. Ein <span class=\"ecbx-1095\">lokales Minimum <\/span>und ein <span class=\"ecbx-1095\">isoliertes lokales Minimum <\/span>wird analog definiert. <\/p><p class=\"indent\">Des                     Weiteren                     sagen                     wir,                     dass <math display=\"inline\"><mi>f<\/mi><\/math> in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein                  <span class=\"ecbx-1095\">lokales                  Extremum                <\/span>annimmt                  und <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> ein                           <span class=\"ecbx-1095\">lokaler                           Extremwert                        <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> ist,                                                                                                                 falls <math display=\"inline\"><mi>f<\/mi><\/math> ein lokales         Minimum         oder         ein         lokales         Maximum         in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> annimmt. <\/p> <\/div> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z6caadac81793\"> <a id=\"x1-229002r17\"><\/a> <span class=\"ecbx-1095\">Proposition 8.17 <\/span>(Notwendige Bedingung f\u00fcr Extremum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge und <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">eine reellwertige Funktion auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nimmt in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein lokales Extremum an, <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbar und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist sowohl ein rechtsseitiger als auch ein linksseitiger H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Die Annahme in Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-229002r17\">8.17<\/a>, dass sich die Menge <math display=\"inline\"><mi>D<\/mi><\/math> dem Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> sowohl von links als auch von rechts n\u00e4hert, ist notwendig, da wir <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> von links und von rechts mit Differenzenquotienten approximieren wollen. In konkreten Rechenbeispielen                                                                                                                                                                           ist sie jedoch meist erf\u00fcllt. Beispielsweise ist dies so in allen Punkten eines Intervalls abgesehen von den Endpunkten erf\u00fcllt. <\/p><p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Ohne Beschr\u00e4nkung der Allgemeinheit nehmen wir an, dass <math display=\"inline\"><mi>f<\/mi><\/math> ein lokales Maximum in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> annimmt (sonst ersetzt man <math display=\"inline\"><mi>f<\/mi><\/math> durch <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math>). Da <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> differenzierbar ist und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> von links und rechts angen\u00e4hert werden kann, existieren sowohl der linksseitige als auch der rechtsseitige Grenzwert der Differenzenquotienten bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und beide sind gleich <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi><\/math> hinreichend nahe bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> gilt und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr die Bewegung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2198<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> erf\u00fcllt ist. Weiters ist aber auch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da wiederum <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi><\/math> hinreichend nahe bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> gilt und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr die Bewegung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2197<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> erf\u00fcllt ist. Unter dem Strich erhalten wir <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Falls der Definitionsbereich <math display=\"inline\"><mi>D<\/mi><\/math> ein Intervall ist, so besagt Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-229002r17\">8.17<\/a> das Folgende. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z74135e7c040b\"> <a id=\"x1-229003r18\"><\/a> <span class=\"ecbx-1095\">Korollar 8.18 <\/span>(Lokale Extremwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">Intervall und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nimmt in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">ein lokales Extremum an. Dann bestehen genau folgende M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeiten:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist ein in <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">enthaltener Endpunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>I<\/mi><\/math><\/span><span class=\"period\">,<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">nicht differenzierbar oder<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbar und <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere sind alle Punkte, wo <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ein lokales Extremum f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine differenzierbare Funktion auf einem offenen Intervall annimmt,<\/span> <span class=\"ecti-1095\">Nullstellen der Ableitung.<\/span> <\/p> <\/div> <p class=\"indent\">Man beachte, dass alle F\u00e4lle in obigem Korollar eintreten k\u00f6nnen (wieso?). Des Weiteren ist die Umkehrung von Proposition&nbsp;<a href=\"..\/..\/chapter\/die-ableitung#x1-229002r17\">8.17<\/a> nicht richtig, wie wir in folgender \u00dcbung zeigen wollen. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb8c0bfd2780b\"> <a id=\"x1-229007r19\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.19.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie alle lokalen Extremwerte des Polynoms <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie alle lokalen Extremwerte der Funktion <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <a id=\"x1-229010r229\"><\/a> <h4 id=\"ze9255863d881\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.4 <\/span> <a id=\"x1-2300004\"><\/a>Stetige Differenzierbarkeit<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge, so dass jeder Punkt in <math display=\"inline\"><mi>D<\/mi><\/math> ein H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> ist (wie zum Beispiel bei einem Intervall mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math>). Falls <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine differenzierbare Funktion ist (also bei jedem Punkt in <math display=\"inline\"><mi>D<\/mi><\/math> differenzierbar ist), k\u00f6nnen wir die Ableitung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">als eine neue Funktion betrachten. Ist <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> stetig, so nennen wir <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">stetig differenzierbar<\/span>. <\/p><p class=\"indent\">Man beachte, dass eine differenzierbare Funktion nicht zwingend stetig differenzierbar sein muss. Wir illustrieren dies in einem Beispiel. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z699679c5534f\"> <a id=\"x1-230001r20\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.20 <\/span>(Unstetige Ableitung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten zu <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die Abbildung <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd> <mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, ist<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">auch bei<\/span> <math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">stetig (siehe das Sandwich<\/span> <span class=\"ecti-1095\">Lemma <\/span><a href=\"#x1-303006r6\"><span class=\"ecti-1095\">B.6<\/span><\/a><span class=\"ecti-1095\">). Die Ableitung von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert und ist durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>p<\/mi><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben. Hierbei verwendeten wir auch, dass die Ableitung von<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gegeben ist. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">die Ableitung von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir keine allgemeine Ableitungsregel verwenden und manipulieren deswegen den<\/span> <span class=\"ecti-1095\">Grenzwert<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>p<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gesetzt haben.<\/span> <span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist, dann existiert<\/span> <span class=\"ecti-1095\">wegen <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>p<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">der Grenzwert<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>p<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">nicht und somit ist<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">nicht differenzierbar.<\/span> <span class=\"ecti-1095\">Wir nehmen nun <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">an, womit <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">(ebenso auf Grund des Sandwich Lemmas) eine rechtsseitige Ableitung<\/span> <math display=\"inline\"><msubsup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">besitzt. Analog k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen<\/span> <span class=\"ecti-1095\">wir den Grenzwert mit <\/span><math display=\"inline\"><msubsup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">berechnen und erhalten drei F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">ist, ist <\/span><math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <mrow> <mo>\u2032<\/mo><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">in jeder Umgebung von <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und insbesondere nicht stetig bei <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Somit ist <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">nicht stetig differenzierbar.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">ist, ist <\/span><math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <mrow> <mo>\u2032<\/mo><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt, aber nicht stetig bei <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">da der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">nicht existiert.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">ist,<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> <span class=\"ecti-1095\">stetig<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist stetig differenzierbar. Man beachte aber, dass<\/span> <math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> <span class=\"ecti-1095\">nicht differenzierbar<\/span> <span class=\"ecti-1095\">sein muss, da f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>3<\/mn><\/math> <span class=\"ecti-1095\">der Grenzwert<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>p<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>p<\/mi><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>p<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>3<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>p<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/mtd><mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd><mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">nicht existiert.<\/span><\/p><\/div><\/div> <\/div> <p class=\"indent\">Das Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-230001r20\">8.20<\/a> l\u00e4sst sich mit fraktalen Konstruktionen stark versch\u00e4rfen. In der Tat kann man eine differenzierbare Funktion auf dem Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> finden, deren Ableitung \u00fcberabz\u00e4hlbar viele Unstetigkeitsstellen besitzt (beispielsweise auf der                                                                                                                                                                           Cantor-Menge). Eine Konstruktion dieser Art finden Sie in Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-2570002\">8.6.2<\/a>. <a id=\"x1-230002r230\"><\/a> <\/p> <h4 id=\"z948dc2235282\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.5 <\/span> <a id=\"x1-2310005\"><\/a>Ableitungen h\u00f6herer Ordnung<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge, so dass jeder Punkt in <math display=\"inline\"><mi>D<\/mi><\/math> ein H\u00e4ufungspunkt ist, und sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine Funktion. Falls <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <\/math> existiert und differenzierbar ist, nennen wir <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">zweimal<\/span> <span class=\"ecbx-1095\">differenzierbar<\/span>. Die Funktion <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> ist die <span class=\"ecbx-1095\">zweite Ableitung <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> und wird auch mit <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> oder <math display=\"inline\"><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/math> bezeichnet. Falls die unabh\u00e4ngige Variable <math display=\"inline\"><mi>t<\/mi><\/math> ist, schreiben wir <math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u1e1f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u22c5<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/math> f\u00fcr die zweite Ableitung nach <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere erhalten wir, dass eine zweimal differenzierbare Funktion <math display=\"inline\"><mi>f<\/mi><\/math> stetig differenzierbar ist. <\/p><p class=\"indent\">Induktiv kann man nun h\u00f6here Differenzierbarkeit und h\u00f6here Ableitungen definieren. Formal definieren wir also die Ableitungen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mspace class=\"quad\" width=\"1em\" \/><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <\/math> f\u00fcr ein <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> (auf ganz <math display=\"inline\"><mi>D<\/mi><\/math>) existiert, heisst <math display=\"inline\"><mi>f<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecbx-1095\">-mal differenzierbar<\/span>. Falls die <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecbx-1095\">-te<\/span> <span class=\"ecbx-1095\">Ableitung <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> zus\u00e4tzlich stetig ist, heisst <math display=\"inline\"><mi>f<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecbx-1095\">-mal stetig differenzierbar<\/span>. Die Menge der <math display=\"inline\"><mi>n<\/mi><\/math>-mal stetig differenzierbaren Funktionen auf <math display=\"inline\"><mi>D<\/mi><\/math> bezeichnen wir mit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><a id=\"dx1-231001\"><\/a> <\/p><p class=\"indent\">F\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> kann man eine Funktion finden, die zwar <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbar, aber nicht <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math>-mal differenzierbar ist. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1c58f798483a\"> <a id=\"x1-231002r21\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.21.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">stetig differenzierbar, aber nicht <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbar ist.<\/span> <\/p> <\/div> <p class=\"indent\">Wir sagen, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">glatt <\/span>oder <span class=\"ecbx-1095\">beliebig oft differenzierbar <\/span>ist, falls <math display=\"inline\"><mi>f<\/mi><\/math> f\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbar ist. Ist <math display=\"inline\"><mi>f<\/mi><\/math> glatt, so sind insbesondere alle Ableitungen von <math display=\"inline\"><mi>f<\/mi><\/math> stetig (<math display=\"inline\"><mi>f<\/mi><\/math> ist also beliebig oft stetig differenzierbar). Die Menge der glatten Funktionen auf <math display=\"inline\"><mi>D<\/mi><\/math> bezeichnen wir mit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>C<\/mi><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir kennen bereits einige Beispiele glatter Funktionen. Dazu geh\u00f6ren die Polynome, da diese nach Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\">8.6<\/a> differenzierbar sind und da deren Ableitung ein Polynom ist, womit die Aussage aus Induktion folgt. Ebenfalls glatt sind die Funktion <math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><\/math> nach Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-228001r3\">8.3<\/a> und \u00dcbung <a href=\"..\/..\/chapter\/die-ableitung#x1-228005r4\">8.4<\/a>. Etwas interessanter, aber nicht ganz unerwartet ist vermutlich folgendes Beispiel. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z8fb880fdd535\"> <a id=\"x1-231003r22\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.22 <\/span>(Logarithmusfunktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Der Logarithmus <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist glatt. In der Tat gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>2<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">oder allgemein <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">!<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was sich mit vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion beweisen l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst.<\/span> <\/p> <\/div> <p class=\"indent\">Ein \u00fcberraschenderes Beispiel einer glatten Funktion ist vielleicht das folgende. <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zcdd1483c9845\"> <a id=\"x1-231004r23\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.23 <\/span>(Glattes Abklingen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist glatt und demnach auch beliebig oft stetig differenzierbar, siehe das folgende Bild.<\/span> <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=566&amp;url=Pictures\/ableitung\/expm1overx.pdf\"><img id=\"ze7309b3f1788\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/ableitung\/expm1overx.svg\" width=\"566\"><\/div>  <\/div> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gibt es nichts zu zeigen, da die Ableitung der Nullfunktion die Nullfunktion ist. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ergibt sich dies mittels Induktion, der Kettenregel (Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\"><span class=\"ecti-1095\">8.8<\/span><\/a><span class=\"ecti-1095\">), Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228015r9\"><span class=\"ecti-1095\">8.9<\/span><\/a><span class=\"ecti-1095\">,<\/span> <span class=\"ecti-1095\">der Produktregel in Proposition <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\"><span class=\"ecti-1095\">8.5<\/span><\/a> <span class=\"ecti-1095\">und Korollar <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\"><span class=\"ecti-1095\">8.6<\/span><\/a><span class=\"ecti-1095\">. In der Tat gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und (da die konkrete Formel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">schnell kompliziert wird) allgemeiner<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-231005r4\" \/><mstyle class=\"maketag\"><mtext>(8.4)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r gewisse Polynome <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und jedes <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">haben wir diese Darstellung der Ableitung bereits bewiesen, wobei<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>4<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/math><span class=\"ecti-1095\">. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Induktionsschritt<\/span> <span class=\"ecti-1095\">nehmen wir<\/span> (<a href=\"..\/..\/chapter\/die-ableitung#x1-231005r4\">8.4<\/a>) <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">an und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei das Polynom <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">als <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlt wurde.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Es bleibt noch zu zeigen, dass <\/span><math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">auch in <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">beliebig oft differenzierbar ist. Dabei k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir nicht auf unsere Ableitungsregeln zur<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckgreifen,<\/span> <span class=\"ecti-1095\">sondern m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen dies direkt mit der Definition der Ableitung <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen. Wir behaupten, dass<\/span> <math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Beweis der Behauptung zeigen wir zuerst, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes Polynom<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-231006r5\" \/><mstyle class=\"maketag\"><mtext>(8.5)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">ist. Auf Grund der Linearit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t des Grenzwerts und da<\/span> <math display=\"inline\"><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gilt, gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt es<\/span> <span class=\"ecti-1095\">zu zeigen, dass <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <span class=\"ecti-1095\">Setzen wir <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so erhalten wir, dass diese Behauptung wiederum zu<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>y<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><msup><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent ist. Dies folgt aber mit dem Sandwich-Lemma aus der Ungleichung<\/span> <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac> <mrow> <mi>y<\/mi><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">(siehe Abschnitt <\/span><a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1650003\"><span class=\"ecti-1095\">6.3<\/span><\/a><span class=\"ecti-1095\">).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Wir zeigen nun <\/span><math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">per Induktion. Verwenden wir<\/span> (<a href=\"..\/..\/chapter\/die-ableitung#x1-231006r5\">8.5<\/a>)<span class=\"ecti-1095\">, so erhalten wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Falls wir bereits <\/span><math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">wissen, dann folgt ebenso<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Wir haben nun also gezeigt, dass alle Ableitungen von<\/span> <math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">auf ganz<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">existieren und<\/span> <span class=\"ecti-1095\">somit ist <\/span><math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">glatt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"ze34dbb788ba5\"> <a id=\"x1-231007r24\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.24 <\/span>(Hutfunktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r beliebige reelle Zahlen <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>d<\/mi><\/math> <span class=\"ecti-1095\">eine glatte Funktion <\/span><math display=\"inline\"><mi>\u03c6<\/mi><\/math> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><mi>\u03c6<\/mi><\/math> <span class=\"ecti-1095\">gleich Null ist ausserhalb des Intervalls <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und gleich <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist auf dem Intervall <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Versuchen Sie zuerst geeignet verschobene und gespiegelte Versionen der Funktion<\/span> <math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">aus Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-231004r23\"><span class=\"ecti-1095\">8.23<\/span><\/a> <span class=\"ecti-1095\">zu  kombinieren.  Zum  Start  k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte  man  beispielsweise  die  Abbildung<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/> <mfrac> <mrow> <mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">+<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">betrachten.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wir wenden uns nun wieder allgemeinen Aussagen im Stile von Abschnitt <a href=\"..\/..\/chapter\/die-ableitung#x1-2280002\">8.1.2<\/a> zu. Aus Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a> l\u00e4sst sich folgendes Korollar deduzieren. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"ze6e556d944e3\"> <a id=\"x1-231008r25\"><\/a> <span class=\"ecbx-1095\">Korollar 8.25 <\/span>(Summen und Produkte bei h\u00f6herer Differenzierbarkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge,<\/span> <span class=\"ecti-1095\">so dass jeder Punkt in <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">ist. Seien <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal differenzierbar.<\/span> <span class=\"ecti-1095\">Dann sind <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">ebenso<\/span> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal differenzierbar<\/span> <span class=\"ecti-1095\">und es gilt <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">sowie<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist jedes skalare Vielfache <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbar und <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Die obige Produktregel f\u00fcr h\u00f6here Ableitungen nennt sich auch <span class=\"ecbx-1095\">Leibniz-Regel<\/span>.                                                                                                                                                                           <\/p><p class=\"indent\">Nat\u00fcrlich sind auch Verkn\u00fcpfungen von <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbaren Funktionen <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbar. Allerdings ist es im Gegensatz zum Produkt deutlich schwerer, hier eine explizite Formel anzugeben. Wir beschr\u00e4nken uns deswegen darauf, nur die Differenzierbarkeit zu formulieren. <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"z194ade77368a\"> <a id=\"x1-231009r26\"><\/a> <span class=\"ecbx-1095\">Korollar 8.26 <\/span>(Verkn\u00fcpfungen und h\u00f6here Differenzierbarkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>E<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Teilmengen, so dass jeder Punkt in <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><mi>E<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><mi>E<\/mi><\/math> <span class=\"ecti-1095\">ist. Sei des Weiteren <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>E<\/mi><\/math> <span class=\"ecti-1095\">eine <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbare Funktion und sei <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>E<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbare Funktion. Dann ist <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbar.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z338913fa3c12\"> <a id=\"x1-231010r27\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.27.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie die Korollare <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-231008r25\"><span class=\"ecti-1095\">8.25<\/span><\/a> <span class=\"ecti-1095\">und <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-231009r26\"><span class=\"ecti-1095\">8.26<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><p class=\"indent\"><\/p><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Die <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-te<\/span> <span class=\"ecti-1095\">Ableitung von <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist eine Linearkombination von Funktionen der Form <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><msub><mrow><mi>k<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><msub><mrow><mi>k<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>k<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>k<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>k<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-231011r226\"><\/a> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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}\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z4c1bcff6261e\" class=\"sectionHead\"><span class=\"titlemark\">8.1 <\/span> <a id=\"x1-2260001\"><\/a>Die Ableitung<\/h3> <a id=\"x1-226001r224\"><\/a> <h4 id=\"z1323e52baf31\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.1 <\/span> <a id=\"x1-2270001\"><\/a>Definition und geometrische Interpretation<\/h4> <p class=\"noindent\">Eine (nicht-vertikale) <span class=\"ecbx-1095\">Gerade <\/span>im <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math> ist eine Teilmenge der Form <math display=\"inline\"> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>m<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> f\u00fcr Parameter <math display=\"inline\"><mi>m<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> oder alternativ ausgedr\u00fcckt der Graph der (affinen) Abbildung <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mi>m<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Meist nennt man Funktionen dieser Form ebenfalls Geraden. Der Parameter <math display=\"inline\"><mi>m<\/mi><\/math> der Geraden <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>m<\/mi><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>q<\/mi><\/math> wird auch die <span class=\"ecbx-1095\">Steigung <\/span>der Geraden genannt. Wir m\u00f6chten uns nun mit Funktionen besch\u00e4ftigen, die sich um einen Punkt im Definitionsbereich durch Geraden approximieren lassen. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1d75e9ac9cb9\"> <a id=\"x1-227001r1\"><\/a> <span class=\"ecbx-1095\">Definition 8.1 <\/span>(Differenzierbarkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge, <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine Funktion und <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein H\u00e4ufungspunkt von <span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecbx-1095\">differenzierbar <\/span>ist, falls der Grenzwert                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-227002r1\" \/><mstyle class=\"maketag\"><mtext>(8.1)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">existiert. In diesem Fall nennen wir <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die <span class=\"ecbx-1095\">Ableitung <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> bei <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><mi>f<\/mi><\/math> bei jedem H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> in <math display=\"inline\"><mi>D<\/mi><\/math> differenzierbar ist, dann sagen wir auch, dass <math display=\"inline\"><mi>f<\/mi><\/math> (auf <math display=\"inline\"><mi>D<\/mi><\/math>) <span class=\"ecbx-1095\">differenzierbar <\/span>ist und nennen die Funktion <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> definiert auf den H\u00e4ufungspunkten von&nbsp;<math display=\"inline\"><mi>D<\/mi><\/math> in&nbsp;<math display=\"inline\"><mi>D<\/mi><\/math> die <span class=\"ecbx-1095\">Ableitung <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Falls <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ein rechtseitiger H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> ist, dann ist <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecbx-1095\">rechtsseitig differenzierbar<\/span>, wenn die <span class=\"ecbx-1095\">rechtsseitige Ableitung<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">existiert. <span class=\"ecbx-1095\">Linksseitige Differenzierbarkeit <\/span>und die <span class=\"ecbx-1095\">linksseitige<\/span> <span class=\"ecbx-1095\">Ableitung<\/span>&nbsp;<math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> werden analog \u00fcber die Bewegung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2197<\/mo> <mi>a<\/mi><\/math> definiert.                                                                                                                                                                           <\/p> <\/div> <p class=\"indent\">Wir nennen <math display=\"inline\"><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>h<\/mi><\/math> im Zusammenhang mit der Definition in (<a href=\"..\/..\/chapter\/die-ableitung#x1-227002r1\">8.1<\/a>) auch das <span class=\"ecbx-1095\">Inkrement des Arguments <\/span>oder der <span class=\"ecbx-1095\">unabh<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">ngigen<\/span> <span class=\"ecbx-1095\">Variablen <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das <span class=\"ecbx-1095\">Inkrement der Funktion<\/span> und <math display=\"inline\"><mfrac><mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>f<\/mi><\/mrow> <mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> den <span class=\"ecbx-1095\">Differenzenquotienten<\/span>. Die Ableitung von <math display=\"inline\"><mi>f<\/mi><\/math> bei <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">,<\/span> welche in dieser Formulierung der Grenzwert des Differenzenquotienten <math display=\"inline\"><mfrac><mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>f<\/mi><\/mrow> <mrow><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> f\u00fcr <math display=\"inline\"><mo class=\"MathClass-bin\">\u25b3<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>0<\/mn><\/math> ist, schreibt man auch als <math display=\"inline\"><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> und nennt dies den <span class=\"ecbx-1095\">Differentialquotienten <\/span>(in der Leibniz-Notation). Weiters nennt man <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mspace class=\"thinspace\" width=\"0.17em\" \/> <mi class=\"qopname\">d<\/mi><mo>  <\/mo> <mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac><\/math> auch die <span class=\"ecbx-1095\">Ableitung<\/span> <span class=\"ecbx-1095\">nach <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><\/math><\/span><span class=\"period\">,<\/span> was vor allem dann n\u00fctzlich ist, wenn <math display=\"inline\"><mi>f<\/mi><\/math> auch von weiteren Parametern abh\u00e4ngen darf. <\/p><p class=\"indent\">Wir m\u00f6chten aber betonen, dass <math display=\"inline\"><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> nicht als Quotient, sondern nur als Grenzwert von Quotienten definiert wurde. Falls die unabh\u00e4ngige Variable <math display=\"inline\"><mi>t<\/mi><\/math> (f\u00fcr Zeit) und nicht <math display=\"inline\"><mi>x<\/mi><\/math> ist, dann verwendet man manchmal auch die Notation <math display=\"inline\"><mi>\u1e8b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>\u1e8f<\/mi> <\/math> f\u00fcr die Ableitung von Funktionen <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Eine weitere Schreibweise der Definition in (<a href=\"..\/..\/chapter\/die-ableitung#x1-227002r1\">8.1<\/a>) ist in der Landau-Notation (siehe Abschnitt <a href=\"..\/..\/chapter\/landau-notation#x1-1820006\">6.6<\/a>) <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>a<\/mi><\/math> oder \u00e4quivalenterweise <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-227003r2\" \/><mstyle class=\"maketag\"><mtext>(8.2)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Hierbei wird die Funktion <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> das <span class=\"ecbx-1095\">Differential <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> genannt und die Gerade <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> die <span class=\"ecbx-1095\">affine <\/span>oder <span class=\"ecbx-1095\">lineare<\/span> <span class=\"ecbx-1095\">Approximation <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> oder die <span class=\"ecbx-1095\">Tangente <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> bei <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">,<\/span> siehe auch Figur&nbsp;<a href=\"..\/..\/chapter\/die-ableitung#x1-227005r1\">8.1<\/a>. Wir erinnern daran, dass wir in (<a href=\"..\/..\/chapter\/die-ableitung#x1-227003r2\">8.2<\/a>) <math display=\"inline\"><mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> als Platzhalter einer Funktion (welcher?) interpretieren, die f\u00fcr <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>a<\/mi><\/math> schneller abf\u00e4llt als <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere ist wegen (<a href=\"..\/..\/chapter\/die-ableitung#x1-227003r2\">8.2<\/a>) <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <math display=\"inline\"><mi>f<\/mi><\/math> ist bei <math display=\"inline\"><mi>a<\/mi><\/math> stetig, wenn <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> differenzierbar ist. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze631e7dd7186\"> <a id=\"x1-227004r2\"><\/a> <span class=\"ecbx-1095\">Applet 8.2 <\/span>(Bewegung der Sekante)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 688px\"><iframe height=\"550px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/N6Q4PScN\/width\/688\/height\/550\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir sehen den Graphen einer Funktion und wie die Sekante zwischen <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><\/math> <span class=\"ecti-1095\">sich bei den meisten Fusspunkten <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">der Tangente bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">n<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hert falls <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>h<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=565&amp;url=Pictures\/ableitung\/def.pdf\"><img decoding=\"async\" id=\"zfbbe15b946f3\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/ableitung\/def.svg\" width=\"565\" \/><\/div> <a id=\"x1-227005r1\"><\/a> <a id=\"x1-227006\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;8.1:              <\/span><span class=\"content\">Die              geometrische              Interpretation               der          Ableitung          einer          reellwertigen          Funktion               <math display=\"inline\"><mi>f<\/mi><\/math>             bei&nbsp;<math display=\"inline\"><mi>a<\/mi><\/math>             ist     die     Steigung     der     Tangenten     des     Graphen     bei               <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span>               Denn wenn <math display=\"inline\"><mi>x<\/mi><\/math>             gegen <math display=\"inline\"><mi>a<\/mi><\/math>             strebt,           wird           die           Sekante,           die           durch               <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math>               und <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math>               geht       und       den       Differenzenquotienten       als       Steigung               besitzt,     immer     mehr     zur     Tangente     des     Graphen     bei               <span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi><\/math><\/span><span class=\"period\">.<\/span> &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <p class=\"indent\">H\u00e4ufig wird in diesem Kapitel (und dem n\u00e4chsten) der Definitionsbereich <math display=\"inline\"><mi>D<\/mi><\/math> der betrachteten                                                                                                                                                                           Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi><mspace class=\"nbsp\" width=\"0.33em\" \/> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ein Intervall <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>I<\/mi><\/math> mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> sein. Dies hat den Vorteil, dass jeder Punkt in <math display=\"inline\"><mi>I<\/mi><\/math> ein H\u00e4ufungspunkt ist (wieso?) und es somit f\u00fcr jeden Punkt in <math display=\"inline\"><mi>I<\/mi><\/math> Sinn macht, nach der Differenzierbarkeit von <math display=\"inline\"><mi>f<\/mi><\/math> bei diesem Punkt zu fragen. Wir wollen dies aber weder in der Definition noch in den zu besprechenden Ableitungsregeln voraussetzen, damit wir beispielsweise auch von der Ableitung der Funktion <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> sprechen k\u00f6nnen. <\/p><p class=\"indent\">Meist werden wir reellwertige Funktionen betrachten. Doch wird es teilweise n\u00fctzlich sein, den Begriff der Ableitung und manche der Gesetze auch f\u00fcr komplexwertige Funktionen verwenden zu k\u00f6nnen. Wir bemerken also, dass Definition <a href=\"..\/..\/chapter\/die-ableitung#x1-227001r1\">8.1<\/a> analog auch f\u00fcr komplexwertige Funktionen verwendet werden kann. Wie in Abschnitt <a href=\"..\/..\/chapter\/folgen-und-konvergenz#x1-1480004\">5.3.4<\/a> l\u00e4uft dies darauf hinaus, dass sowohl Real- als auch Imagin\u00e4rteil differenzierbar sein sollten. <\/p><p class=\"indent\">Schlussendlich wollen wir noch anmerken, dass die Ableitung eine rein lokale Operation darstellt. Genauer gesagt, angenommen <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> ist ein H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> und <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> sind bei&nbsp;<math display=\"inline\"><mi>a<\/mi><\/math> differenzierbare Funktionen, so dass es ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt mit <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dies ergibt sich unmittelbar aus der Definition der Grenzwerte, die <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> definieren (wieso?). Wir werden dies im Folgenden teils implizit verwenden. <a id=\"x1-227007r227\"><\/a> <\/p> <h4 id=\"za345598f62ae\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.2 <\/span> <a id=\"x1-2280002\"><\/a>Beispiele und Ableitungsregeln<\/h4> <p class=\"noindent\">Wir wollen nun zeigen, dass viele der uns gel\u00e4ufigen Funktionen differenzierbar sind und dass wir die Ableitung (meistens) mittels einigen konkreten Gesetzen bestimmen k\u00f6nnen. Wir beginnen aber zuerst mit elementaren Beispielen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb2f92347a7d0\"> <a id=\"x1-228001r3\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.3 <\/span>(Erste Beispiele differenzierbarer Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Konstante  Funktionen  sind  <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berall  differenzierbar  und  haben  die  Nullfunktion  als<\/span> <span class=\"ecti-1095\">Ableitung (wieso?).<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">tsfunktion <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist differenzierbar und ihre Ableitung ist die konstante<\/span> <math display=\"inline\"><mn>1<\/mn><\/math><span class=\"ecti-1095\">-Funktion,<\/span> <span class=\"ecti-1095\">denn<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Exponentialfunktion <\/span><math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist differenzierbar und ihre Ableitung ist die Exponentialfunktion. Allgemeiner behaupten wir, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">ein festes<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">(oder<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><span class=\"ecti-1095\">) die<\/span> <span class=\"ecti-1095\">Ableitung von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">durch<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gegeben<\/span><span class=\"ecti-1095\">&nbsp;ist. In<\/span> <span class=\"ecti-1095\">der Tat gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>1<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>k<\/mi><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msup><msup><mrow><mi>h<\/mi><\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/munderover> <mfrac><mrow><msup><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>h<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>\u03b1<\/mi><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>\u03b1<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">da die Abbildung <\/span><math display=\"inline\"><mi>h<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msubsup><mrow><mi class=\"MathClass-op\">\u2211<\/mi><mo> <\/mo> <\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>\u221e<\/mi><\/mrow><\/msubsup> <mfrac><mrow><msup><mrow><mi>\u03b1<\/mi><\/mrow><mrow><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u2113<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">!<\/mo><\/mrow><\/mfrac><msup><mrow><mi>h<\/mi><\/mrow><mrow><mi>\u2113<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">nach Satz <\/span><a href=\"..\/..\/chapter\/potenzreihen#x1-200002r56\"><span class=\"ecti-1095\">7.56<\/span><\/a> <span class=\"ecti-1095\">stetig ist.<\/span><\/p><\/dd><\/dl> <\/div> <p class=\"indent\">Wir besprechen weitere Beispiele von differenzierbaren Funktionen und ein Beispiel einer nicht-differenzierbaren Funktion in der folgenden \u00dcbung. <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4f9bd5b67a64\"> <a id=\"x1-228005r4\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.4 <\/span>(Weitere differenzierbare Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Verwenden Sie die Additionstheoreme aus Abschnitt <\/span><a href=\"..\/..\/chapter\/trigonometrische-funktionen#x1-2090001\"><span class=\"ecti-1095\">7.6.1<\/span><\/a> <span class=\"ecti-1095\">(oder Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228001r3\"><span class=\"ecti-1095\">8.3<\/span><\/a> <span class=\"ecti-1095\">(iii)), um zu<\/span> <span class=\"ecti-1095\">zeigen, dass der Sinus und der Kosinus differenzierbare Funktionen sind und die<\/span> <span class=\"ecti-1095\">Ableitungsregeln<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">gelten.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Funktionen <\/span><math display=\"inline\"><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo> <\/math> <span class=\"ecti-1095\">differenzierbar sind und verifizieren Sie die Ableitungsregeln<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iv)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Zeigen Sie, dass die Betragsfunktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">nicht differenzierbar ist und bestimmen Sie bei jedem Punkt in<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">die linksseitige und die rechtsseitige Ableitung.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\">Wie in (ii) und (iii) von \u00dcbung <a href=\"..\/..\/chapter\/die-ableitung#x1-228005r4\">8.4<\/a> schon verwendet, wollen wir f\u00fcr Funktionen wie zum Beispiel die Funktion <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> die durch Formeln gegeben sind, nicht immer einen Namen einf\u00fchren, um die Ableitung hinschreiben zu k\u00f6nnen. Stattdessen schreiben wir <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und meinen damit, dass die Funktion <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/math> auf ihrem maximalen Definitionsbereich differenzierbar ist und dass ihre Ableitung bei <math display=\"inline\"><mi>x<\/mi><\/math> durch <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> gegeben ist. Insbesondere ist&nbsp;<math display=\"inline\"><mi>x<\/mi><\/math> in obiger Gleichung nicht als Zahl, sondern vielmehr als Argument der Funktion und der Ableitung zu erachten. <\/p><p class=\"indent\">Wie schon bei stetigen und Riemann-integrierbaren Funktionen m\u00f6chten wir nicht immer von Hand zeigen m\u00fcssen, dass eine gegebene Funktion differenzierbar ist. Stattdessen wollen wir allgemeine Regeln beweisen, auf die sich die Differenzierbarkeit verschiedener Funktionen zur\u00fcckf\u00fchren l\u00e4sst. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4bd25f5e8523\"> <a id=\"x1-228010r5\"><\/a> <span class=\"ecbx-1095\">Proposition 8.5 <\/span>(Summen und Produkte differenzierbarer Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">Teilmenge und <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bei<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar.<\/span> <span class=\"ecti-1095\">Dann sind <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">bei<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist jedes skalare Vielfache von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dies gilt ebenso f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r komplexwertige Funktionen.<\/span> <\/p> <\/div> <p class=\"indent\">Somit bilden die bei <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> differenzierbaren reellwertigen Funktionen einen Unterraum des Vektorraums <math display=\"inline\"><mi mathvariant=\"bold-script\">\u2131<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> der reellwertigen Funktionen von <math display=\"inline\"><mi>D<\/mi><\/math> nach <math display=\"inline\"><mi>\u211d<\/mi><\/math> und die Ableitung bei <math display=\"inline\"><mi>a<\/mi><\/math> ist eine lineare Abbildung von diesem Unterraum nach <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Ableitungsregel f\u00fcr das Produkt zweier Funktionen wird auch die <span class=\"ecbx-1095\">Produktregel<\/span> genannt. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir berechnen unter Verwendung der Eigenschaften des Grenzwerts in Abschnitt&nbsp;<a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-1750001\">6.4.1<\/a> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>a<\/mi><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mi>g<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>a<\/mi><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da <math display=\"inline\"><mi>g<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> stetig ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z92b4e6b65f0e\"> <a id=\"x1-228011r6\"><\/a> <span class=\"ecbx-1095\">Korollar 8.6 <\/span>(Differenzierbarkeit von Polynomen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Reelle Polynome sind auf ganz <\/span><math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-228012r3\" \/><mstyle class=\"maketag\"><mtext>(8.3)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Nach Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a> und Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\">8.6<\/a> ist insbesondere die Ableitung eines Polynoms wieder ein Polynom. Weiters ist <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> eine lineare Abbildung. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis von Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\">8.6<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Die F\u00e4lle <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> wurden bereits in Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-228001r3\">8.3<\/a> besprochen. Wir beweisen (<a href=\"..\/..\/chapter\/die-ableitung#x1-228012r3\">8.3<\/a>) per Induktion nach <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi><\/math><\/span><span class=\"period\">.<\/span> Angenommen f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>n<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">.<\/span> Dann folgt aus Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a>, dass <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/math> differenzierbar ist und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">erf\u00fcllt, was den Induktionsbeweis abschliesst. Differenzierbarkeit eines beliebigen Polynoms folgt nun aus der Linearit\u00e4t der Ableitung in Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd918db443989\"> <a id=\"x1-228013r7\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.7 <\/span>(Potenzregel mittels Binomialsatz)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie Korollar <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\"><span class=\"ecti-1095\">8.6<\/span><\/a> <span class=\"ecti-1095\">direkt unter Verwendung des Binomialsatzes. Beweisen Sie des<\/span> <span class=\"ecti-1095\">Weiteren,              dass              der              Kern              der              Abbildung<\/span> <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-open\">[<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math> <span class=\"ecti-1095\">aus den konstanten Polynomen besteht. Sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter werden wir sehen, dass nicht nur Polynome<\/span> <span class=\"ecti-1095\">mit Ableitung      Null,      sondern      auch      differenzierbare      Funktionen      auf<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">mit Ableitung Null konstant sein m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen.<\/span> <\/p> <\/div> <p class=\"indent\">Wieder in Analogie zur Diskussion von stetigen Funktionen (genauer Proposition <a href=\"..\/..\/chapter\/stetigkeit#x1-94011r52\">3.52<\/a>) wollen wir zeigen, dass die Verkn\u00fcpfung zweier differenzierbaren Funktionen auch differenzierbar&nbsp;ist. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z0ea6b008f69c\"> <a id=\"x1-228014r8\"><\/a> <span class=\"ecbx-1095\">Satz 8.8 <\/span>(Kettenregel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>E<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Teilmengen<\/span> <span class=\"ecti-1095\">und sei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt. Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>E<\/mi><\/math> <span class=\"ecti-1095\">eine bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbare<\/span> <span class=\"ecti-1095\">Funktion, so dass <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><math display=\"inline\"><mi>E<\/mi><\/math> <span class=\"ecti-1095\">ist, und sei <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>E<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine<\/span> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbare<\/span> <span class=\"ecti-1095\">Funktion. Dann ist <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbar und<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Wir bemerken, dass man zwar versucht sein mag, f\u00fcr den Beweis der Kettenregel den Differenzenquotienten <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">mit <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> zu erweitern. Dies ist im Allgemeinen aber nicht erlaubt, da wir nicht ausschliessen k\u00f6nnen, dass <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr gewisse Punkte <math display=\"inline\"><mi>x<\/mi><\/math> nahe bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ist. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir verwenden stattdessen die Umformulierung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mi>o<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> oder genauer formuliert <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei die Funktion <math display=\"inline\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><\/math> auf <math display=\"inline\"><mi>D<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> gegeben ist und bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> stetig ist. Ebenso gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wobei die bei <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> stetige Funktion <math display=\"inline\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><\/math> auf <math display=\"inline\"><mi>E<\/mi><\/math> durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"left\"><mfrac><mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>y<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>E<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"left\"><mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>E<\/mi><\/math> gegeben ist. Zusammen ergibt sich durch Einsetzen von <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/> <mo class=\"MathClass-bin\">+<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">,<\/span> womit <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><\/mtd> <mtd class=\"align-even\"><mfrac><mrow><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>g<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>\ud835\udf00<\/mi><\/mrow><mrow><mi>f<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">wie gew\u00fcnscht. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Abgesehen von Summen, Produkten und Verkn\u00fcpfungen von differenzierbaren Funktionen, m\u00f6chten wir zeigen, dass Quotienten von differenzierbaren Funktionen differenzierbar sind. Wir beginnen dazu mit einem wichtigen Beispiel. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zfbc6700c3814\"> <a id=\"x1-228015r9\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.9 <\/span>(Kehrwert)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math><span class=\"ecti-1095\">. Dann ist<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar<\/span> <span class=\"ecti-1095\">und es gilt <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">In der Tat ist<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><mo class=\"MathClass-bin\">+<\/mo><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/mrow> <mrow><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><mi>h<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>h<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wegen der Stetigkeit von <\/span><math display=\"inline\"><mi>h<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>h<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>x<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z4ca95e2cf962\"> <a id=\"x1-228016r10\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.10 <\/span>(Negative Potenzen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Berechnen Sie <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Unter Kombination der Kettenregel und Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-228015r9\">8.9<\/a> erh\u00e4lt man nun folgendes Korollar. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd90a013a1ed5\"> <a id=\"x1-228017r11\"><\/a> <span class=\"ecbx-1095\">Korollar 8.11 <\/span>(Quotientenregel)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge,<\/span> <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt<\/span> <span class=\"ecti-1095\">und seien <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">bei<\/span> <math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar.<\/span> <span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, dann<\/span> <span class=\"ecti-1095\">ist auch <\/span><math display=\"inline\"><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>a<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>g<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Man beachte, dass der (nat\u00fcrliche) Definitionsbereich der Funktion <span class=\"maperiod\"><math display=\"inline\"><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> der in obigem Korollar nicht erw\u00e4hnt wurde, die Teilmenge <math display=\"inline\"><mi>E<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> ist. Da <math display=\"inline\"><mi>g<\/mi><\/math> beim Punkt <math display=\"inline\"><mi>a<\/mi><\/math> differenzierbar ist, ist <math display=\"inline\"><mi>g<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> stetig. Insbesondere ist, da <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> ist, <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> f\u00fcr alle&nbsp;<math display=\"inline\"><mi>x<\/mi><\/math> nahe genug bei <math display=\"inline\"><mi>a<\/mi><\/math> und <math display=\"inline\"><mi>a<\/mi><\/math> ist ein H\u00e4ufungspunkt von <span class=\"maperiod\"><math display=\"inline\"><mi>E<\/mi><\/math><\/span><span class=\"period\">.<\/span> Damit macht es auch Sinn, von Differenzierbarkeit von <math display=\"inline\"><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> zu sprechen. <\/p><p class=\"indent\">Eine direkte Konsequenz von Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228017r11\">8.11<\/a> ist, dass rationale Funktionen differenzierbar sind, wo definiert. Wir erinnern daran, dass eine rationale Funktion eine Funktion der Form <math display=\"inline\"><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <\/math> ist, wobei <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> reelle Polynome sind und <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> nicht das Nullpolynom ist. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis von Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228017r11\">8.11<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Es bezeichne <math display=\"inline\"><mi>\u03c8<\/mi><\/math> die Funktion <span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> welche nach Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-228015r9\">8.9<\/a> differenzierbar ist. Wir kombinieren dies mit der Kettenregel (Satz <a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\">8.8<\/a>) und erhalten,                                                                                                                                                                           dass die Funktion <math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03c8<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>g<\/mi><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> differenzierbar ist mit Ableitung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Verwenden wir nun die Produktregel in Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a>, so ergibt sich, dass <math display=\"inline\"><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac><\/math> bei <math display=\"inline\"><mi>a<\/mi><\/math> differenzierbar ist und <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>f<\/mi><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>a<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>g<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>g<\/mi><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">erf\u00fcllt, was zu zeigen war. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Die Kettenregel erlaubt uns die Berechnung der Ableitung von beliebig kompliziert anmutenden konkreten Beispielen, wobei man stur von aussen nach innen vorgeht wie in folgendem Beispiel. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z91d4528b7bed\"> <a id=\"x1-228018r12\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.12 <\/span>(Vierfach verschachtelte Funktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir bestimmen die Ableitung der Funktion<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">mittels mehrmaligem Anwenden der Kettenregel (Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\"><span class=\"ecti-1095\">8.8<\/span><\/a><span class=\"ecti-1095\">). Da<\/span> <math display=\"inline\"><msup><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <\/math> <span class=\"ecti-1095\">erhalten wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Ebenso<\/span> <span class=\"ecti-1095\">ist wegen <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><\/math> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>h<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><msup><mrow><mi>h<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei <\/span><math display=\"inline\"><mi>h<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>h<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mn>2<\/mn><mi>x<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dadurch erhalten wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><mn>2<\/mn><mi>x<\/mi><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z697f483756fa\"> <a id=\"x1-228019r13\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.13 <\/span>(Nochmals vierfach verschachtelt)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Bestimmen Sie die Ableitung von der Funktion <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">cos<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">sin<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Unsere vorl\u00e4ufig letzte allgemeine Ableitungsregel betrifft die Ableitung der Umkehrabbildung (siehe dazu auch Satz <a href=\"..\/..\/chapter\/der-satz-ueber-die-umkehrabbildung#x1-97001r64\">3.64<\/a> \u00fcber die Existenz einer stetigen Umkehrabbildung). <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z236d51b630a5\"> <a id=\"x1-228020r14\"><\/a> <span class=\"ecbx-1095\">Satz 8.14 <\/span>(Differenzierbarkeit der inversen Funktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>E<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Teilmengen und sei <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>E<\/mi><\/math> <span class=\"ecti-1095\">eine stetige, bijektive Abbildung, deren inverse Abbildung<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-punc\">:<\/mo> <mi>E<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ebenfalls stetig<\/span> <span class=\"ecti-1095\">ist. Falls <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">in dem<\/span> <span class=\"ecti-1095\">H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">differenzierbar ist und <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math> <span class=\"ecti-1095\">gilt, dann ist <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">in <\/span><math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=424&amp;url=Pictures\/ableitung\/invfctder.pdf\"><img decoding=\"async\" id=\"z49d736d8a7f3\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/ableitung\/invfctder.svg\" width=\"424\" \/><\/div> <a id=\"x1-228021r2\"><\/a> <a id=\"x1-228022\"><\/a> <br \/><div class=\"caption\"><span class=\"id\">&nbsp;&nbsp;&nbsp;&nbsp;              Figur&nbsp;8.2:           <\/span><span class=\"content\">Eine           intuitive           Darstellung           von               Satz        <a href=\"..\/..\/chapter\/die-ableitung#x1-228020r14\">8.14<\/a>.        Spiegelt        man        den        Graphen        von               <math display=\"inline\"><mi>f<\/mi><\/math>             und die Tangente beim Punkt <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math>               um die Gerade <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/math>             in <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span>               so erh\u00e4lt man den Graphen von <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/math>                   und,      das      ist      die      Behauptung,      die      Tangente      bei               <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span>               Eine          kurze          Rechnung          zeigt,          dass          die               Spiegelung            einer            Gerade            mit            Steigung               <math display=\"inline\"><mi>m<\/mi><\/math>             um <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>y<\/mi><\/math>             Steigung <math display=\"inline\"> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>m<\/mi><\/mrow><\/mfrac><\/math>                hat.                                                                                        &nbsp;&nbsp;&nbsp;&nbsp; <\/span><\/div> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir bemerken zuerst, dass <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> ein H\u00e4ufungspunkt von <math display=\"inline\"><mi>E<\/mi><\/math> ist, womit man von Differenzierbarkeit bei <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> sprechen darf. Tats\u00e4chlich ist nach Annahme <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein H\u00e4ufungspunkt und es existiert eine Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> mit <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math><\/span><span class=\"period\">.<\/span> Da <math display=\"inline\"><mi>f<\/mi><\/math> stetig ist, gilt <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u221e<\/mi><\/math> und da <math display=\"inline\"><mi>f<\/mi><\/math> bijektiv ist, gilt <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Sei nun <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> eine Folge in <span class=\"maperiod\"><math display=\"inline\"><mi>E<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">,<\/span> die gegen <math display=\"inline\"><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> konvergiert. Dann strebt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> in <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gegen <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math><\/span><span class=\"period\">,<\/span> da <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <\/mrow> <\/msup> <\/math> per Annahme stetig ist, und es gilt                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mfrac><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow> <mrow><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">nach der Charakterisierung der Konvergenz einer Funktion mittels Folgen in Lemma <a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\">6.40<\/a>. Da dies aber f\u00fcr jede Folge <math display=\"inline\"><msub><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> wie oben gilt, folgt der Satz wiederum aus Lemma <a href=\"..\/..\/chapter\/grenzwerte-von-funktionen#x1-175007r40\">6.40<\/a>. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z481f18f48603\"> <a id=\"x1-228023r15\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.15 <\/span>(Differenzierbarkeit des Logarithmus und der Potenzfunktionen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-rel\">|<\/mo><mi>y<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist<\/span> <span class=\"ecti-1095\">differenzierbar mit Ableitung <\/span><math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">gegeben durch <\/span><math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle<\/span> <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-bin\">\u2216<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>0<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><span class=\"ecti-1095\">. Denn die Abbildung<\/span> <math display=\"inline\"><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">:<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist die Umkehrabbildung<\/span> <span class=\"ecti-1095\">von <\/span><math display=\"inline\"><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo>  <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und damit folgt<\/span> <span class=\"ecti-1095\">aus Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228020r14\"><span class=\"ecti-1095\">8.14<\/span><\/a><span class=\"ecti-1095\">, dass <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">bei allen Punkten <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">differenzierbar ist mit <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">wobei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><msup><mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo>  <\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">folgt nun<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist<\/span> <math display=\"inline\"><mi>g<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">. Also folgt<\/span> <span class=\"ecti-1095\">Differenzierbarkeit von <\/span><math display=\"inline\"><mi>g<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>y<\/mi><\/math> <span class=\"ecti-1095\">sowie<\/span> <span class=\"ecti-1095\">die Formel <\/span><math display=\"inline\"><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mi class=\"qopname\">log<\/mi><mo>  <\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>y<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">aus der Kettenregel (Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\"><span class=\"ecti-1095\">8.8<\/span><\/a><span class=\"ecti-1095\">).<\/span> <\/p><\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein beliebiges <\/span><math display=\"inline\"><mi>s<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist die Abbildung <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msub><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-rel\">&gt;<\/mo><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">differenzierbar und es gilt<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">In der Tat gilt <\/span><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">per Definition beliebiger Potenzen in Abschnitt <\/span><a href=\"..\/..\/chapter\/die-(komplexe)-exponentialabbildung#x1-2050002\"><span class=\"ecti-1095\">7.5.2<\/span><\/a><span class=\"ecti-1095\">. Aus Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228001r3\"><span class=\"ecti-1095\">8.3<\/span><\/a> <span class=\"ecti-1095\">und der Ableitung der<\/span> <span class=\"ecti-1095\">Logarithmusabbildung folgt somit<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>s<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>s<\/mi><mi class=\"qopname\">log<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>s<\/mi><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><\/mrow><\/msup><mi>s<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>s<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>s<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/p><\/dd><\/dl> <\/div> <a id=\"x1-228026r228\"><\/a> <h4 id=\"z2b16b3834835\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.3 <\/span> <a id=\"x1-2290003\"><\/a>Extremwerte<\/h4> <p class=\"noindent\">Wie wir in diesem Abschnitt sehen werden, ist die Ableitung auch n\u00fctzlich, um Punkte zu finden, bei denen eine Funktion <math display=\"inline\"><mi>f<\/mi><\/math> ihre Maxima und ihre Minima annimmt. Genauer kann man damit die in folgender Definition eingef\u00fchrten Punkte finden. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z164edf7ef666\"> <a id=\"x1-229001r16\"><\/a> <span class=\"ecbx-1095\">Definition 8.16 <\/span>(Lokale Extremwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge und <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir sagen, dass eine Funktion <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> ein <span class=\"ecbx-1095\">lokales Maximum <\/span>in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> annimmt, falls es eine Umgebung <math display=\"inline\"><mi>U<\/mi><\/math> von <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> in <math display=\"inline\"><mi>D<\/mi><\/math> gibt, auf der <math display=\"inline\"><mi>f<\/mi><\/math> durch <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> beschr\u00e4nkt ist. Genauer formuliert heisst dies, dass es ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt, so dass f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> gilt <span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Falls es sogar ein <math display=\"inline\"><mi>\u03b4<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> gibt, so dass <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-bin\">\u2229<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <mi>\u03b4<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2216<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math> gilt, dann nimmt <math display=\"inline\"><mi>f<\/mi><\/math> in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein <span class=\"ecbx-1095\">isoliertes lokales Maximum <\/span>an. Der Wert <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math>                                                                                                                                                                           wird auch ein <span class=\"ecbx-1095\">lokales Maximum <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> genannt. Ein <span class=\"ecbx-1095\">lokales Minimum <\/span>und ein <span class=\"ecbx-1095\">isoliertes lokales Minimum <\/span>wird analog definiert. <\/p><p class=\"indent\">Des                     Weiteren                     sagen                     wir,                     dass <math display=\"inline\"><mi>f<\/mi><\/math> in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> ein                  <span class=\"ecbx-1095\">lokales                  Extremum                <\/span>annimmt                  und <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> ein                           <span class=\"ecbx-1095\">lokaler                           Extremwert                        <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> ist,                                                                                                                 falls <math display=\"inline\"><mi>f<\/mi><\/math> ein lokales         Minimum         oder         ein         lokales         Maximum         in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> annimmt. <\/p> <\/div> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6caadac81793\"> <a id=\"x1-229002r17\"><\/a> <span class=\"ecbx-1095\">Proposition 8.17 <\/span>(Notwendige Bedingung f\u00fcr Extremum)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge und <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">eine reellwertige Funktion auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nimmt in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein lokales Extremum an, <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbar und <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">ist sowohl ein rechtsseitiger als auch ein linksseitiger H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>D<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Dann gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Die Annahme in Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-229002r17\">8.17<\/a>, dass sich die Menge <math display=\"inline\"><mi>D<\/mi><\/math> dem Punkt <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> sowohl von links als auch von rechts n\u00e4hert, ist notwendig, da wir <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/math> von links und von rechts mit Differenzenquotienten approximieren wollen. In konkreten Rechenbeispielen                                                                                                                                                                           ist sie jedoch meist erf\u00fcllt. Beispielsweise ist dies so in allen Punkten eines Intervalls abgesehen von den Endpunkten erf\u00fcllt. <\/p><div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Ohne Beschr\u00e4nkung der Allgemeinheit nehmen wir an, dass <math display=\"inline\"><mi>f<\/mi><\/math> ein lokales Maximum in <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><\/math> annimmt (sonst ersetzt man <math display=\"inline\"><mi>f<\/mi><\/math> durch <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><\/math>). Da <math display=\"inline\"><mi>f<\/mi><\/math> bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> differenzierbar ist und <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> von links und rechts angen\u00e4hert werden kann, existieren sowohl der linksseitige als auch der rechtsseitige Grenzwert der Differenzenquotienten bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> und beide sind gleich <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> Dann ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi><\/math> hinreichend nahe bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> gilt und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr die Bewegung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2198<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/math> erf\u00fcllt ist. Weiters ist aber auch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><msub><mrow><mi>x<\/mi><\/mrow><mrow> <mn>0<\/mn><\/mrow><\/msub><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2197<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/munder><mfrac><mrow><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">da wiederum <math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2264<\/mo> <mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/math> f\u00fcr alle <math display=\"inline\"><mi>x<\/mi><\/math> hinreichend nahe bei <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> gilt und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> f\u00fcr die Bewegung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2197<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> erf\u00fcllt ist. Unter dem Strich erhalten wir <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <p class=\"indent\">Falls der Definitionsbereich <math display=\"inline\"><mi>D<\/mi><\/math> ein Intervall ist, so besagt Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-229002r17\">8.17<\/a> das Folgende. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z74135e7c040b\"> <a id=\"x1-229003r18\"><\/a> <span class=\"ecbx-1095\">Korollar 8.18 <\/span>(Lokale Extremwerte)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>I<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ein<\/span> <span class=\"ecti-1095\">Intervall und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>I<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Angenommen <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">nimmt in <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>I<\/mi><\/math> <span class=\"ecti-1095\">ein lokales Extremum an. Dann bestehen genau folgende M<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">glichkeiten:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist ein in <\/span><math display=\"inline\"><mi>I<\/mi><\/math> <span class=\"ecti-1095\">enthaltener Endpunkt von <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>I<\/mi><\/math><\/span><span class=\"period\">,<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">nicht differenzierbar oder<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist bei <\/span><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">differenzierbar und <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere sind alle Punkte, wo <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">ein lokales Extremum f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r eine differenzierbare Funktion auf einem offenen Intervall annimmt,<\/span> <span class=\"ecti-1095\">Nullstellen der Ableitung.<\/span> <\/p> <\/div> <p class=\"indent\">Man beachte, dass alle F\u00e4lle in obigem Korollar eintreten k\u00f6nnen (wieso?). Des Weiteren ist die Umkehrung von Proposition&nbsp;<a href=\"..\/..\/chapter\/die-ableitung#x1-229002r17\">8.17<\/a> nicht richtig, wie wir in folgender \u00dcbung zeigen wollen. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb8c0bfd2780b\"> <a id=\"x1-229007r19\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.19.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(a)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie alle lokalen Extremwerte des Polynoms <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>x<\/mi><\/math> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(b)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">Finden Sie alle lokalen Extremwerte der Funktion <\/span><math display=\"inline\"><mo class=\"MathClass-rel\">|<\/mo><mi>f<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>3<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <a id=\"x1-229010r229\"><\/a> <h4 id=\"ze9255863d881\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.4 <\/span> <a id=\"x1-2300004\"><\/a>Stetige Differenzierbarkeit<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge, so dass jeder Punkt in <math display=\"inline\"><mi>D<\/mi><\/math> ein H\u00e4ufungspunkt von <math display=\"inline\"><mi>D<\/mi><\/math> ist (wie zum Beispiel bei einem Intervall mit Endpunkten <math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi><\/math> in <math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>\u211d<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math>). Falls <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine differenzierbare Funktion ist (also bei jedem Punkt in <math display=\"inline\"><mi>D<\/mi><\/math> differenzierbar ist), k\u00f6nnen wir die Ableitung                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-punc\">:<\/mo> <mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">als eine neue Funktion betrachten. Ist <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> stetig, so nennen wir <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">stetig differenzierbar<\/span>. <\/p><p class=\"indent\">Man beachte, dass eine differenzierbare Funktion nicht zwingend stetig differenzierbar sein muss. Wir illustrieren dies in einem Beispiel. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z699679c5534f\"> <a id=\"x1-230001r20\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.20 <\/span>(Unstetige Ableitung)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten zu <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">die Abbildung <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><msup><mrow><mo class=\"MathClass-rel\">|<\/mo><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mtd><mtd class=\"array\" columnalign=\"center\"> <mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn> <\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd> <mtd class=\"array\" columnalign=\"center\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle> <mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Da <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ist, ist<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">auch bei<\/span> <math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">stetig (siehe das Sandwich<\/span> <span class=\"ecti-1095\">Lemma <\/span><a href=\"#x1-303006r6\"><span class=\"ecti-1095\">B.6<\/span><\/a><span class=\"ecti-1095\">). Die Ableitung von<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">existiert und ist durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mi>p<\/mi><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo><msup><mrow><mrow><mo fence=\"true\" form=\"prefix\"> |<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">|<\/mo><\/mrow><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">gegeben. Hierbei verwendeten wir auch, dass die Ableitung von<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><\/math> <span class=\"ecti-1095\">durch<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u00d7<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u21a6<\/mo> <mi class=\"qopname\"> sgn<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gegeben ist. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">die Ableitung von <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">bei <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir keine allgemeine Ableitungsregel verwenden und manipulieren deswegen den<\/span> <span class=\"ecti-1095\">Grenzwert<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>p<\/mi><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei wir <\/span><math display=\"inline\"><mi>t<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">gesetzt haben.<\/span> <span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist, dann existiert<\/span> <span class=\"ecti-1095\">wegen <\/span><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>p<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">der Grenzwert<\/span> <math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mi class=\"qopname\">sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>p<\/mi><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">nicht und somit ist<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">nicht differenzierbar.<\/span> <span class=\"ecti-1095\">Wir nehmen nun <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">an, womit <\/span><math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecti-1095\">(ebenso auf Grund des Sandwich Lemmas) eine rechtsseitige Ableitung<\/span> <math display=\"inline\"><msubsup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">+<\/mo> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">besitzt. Analog k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen<\/span> <span class=\"ecti-1095\">wir den Grenzwert mit <\/span><math display=\"inline\"><msubsup><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">berechnen und erhalten drei F<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">lle.<\/span> <\/p> <div class=\"custom-itemize\"><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">ist, ist <\/span><math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <mrow> <mo>\u2032<\/mo><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">in jeder Umgebung von <\/span><math display=\"inline\"><mn>0<\/mn><\/math> <span class=\"ecti-1095\">unbeschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt und insbesondere nicht stetig bei <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Somit ist <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">nicht stetig differenzierbar.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">ist, ist <\/span><math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <mrow> <mo>\u2032<\/mo><\/mrow><\/msubsup><\/math> <span class=\"ecti-1095\">beschr<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">nkt, aber nicht stetig bei <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">da der Grenzwert <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">nicht existiert.<\/span> <\/div><div class=\"item-head\"> <span class=\"tcrm-1095\">\u2022<\/span><\/div><div class=\"item-content\"><span class=\"ecti-1095\">Falls <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">ist,<\/span> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> <span class=\"ecti-1095\">stetig<\/span> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <\/msub> <\/math> <span class=\"ecti-1095\">ist stetig differenzierbar. Man beachte aber, dass<\/span> <math display=\"inline\"><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi> <\/mrow> <mrow> <mo>\u2032<\/mo> <\/mrow> <\/msubsup><\/math> <span class=\"ecti-1095\">nicht differenzierbar<\/span> <span class=\"ecti-1095\">sein muss, da f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>p<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>3<\/mn><\/math> <span class=\"ecti-1095\">der Grenzwert<\/span> <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>p<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>p<\/mi><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>p<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>2<\/mn><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>t<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mi>p<\/mi><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>p<\/mi><\/mrow><\/msup><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>3<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>p<\/mi><\/mrow><\/msup><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>t<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><\/mtd><mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd><mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">nicht existiert.<\/span><\/p><\/div><\/div> <\/div> <p class=\"indent\">Das Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-230001r20\">8.20<\/a> l\u00e4sst sich mit fraktalen Konstruktionen stark versch\u00e4rfen. In der Tat kann man eine differenzierbare Funktion auf dem Intervall <math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">]<\/mo><\/math> finden, deren Ableitung \u00fcberabz\u00e4hlbar viele Unstetigkeitsstellen besitzt (beispielsweise auf der                                                                                                                                                                           Cantor-Menge). Eine Konstruktion dieser Art finden Sie in Abschnitt <a href=\"..\/..\/chapter\/weitere-lernmaterialien#x1-2570002\">8.6.2<\/a>. <a id=\"x1-230002r230\"><\/a> <\/p> <h4 id=\"z948dc2235282\" class=\"subsectionHead\"><span class=\"titlemark\">8.1.5 <\/span> <a id=\"x1-2310005\"><\/a>Ableitungen h\u00f6herer Ordnung<\/h4> <p class=\"noindent\">Sei <math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> eine Teilmenge, so dass jeder Punkt in <math display=\"inline\"><mi>D<\/mi><\/math> ein H\u00e4ufungspunkt ist, und sei <math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> eine Funktion. Falls <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo> <\/mrow> <\/msup> <\/math> existiert und differenzierbar ist, nennen wir <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">zweimal<\/span> <span class=\"ecbx-1095\">differenzierbar<\/span>. Die Funktion <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/math> ist die <span class=\"ecbx-1095\">zweite Ableitung <\/span>von <math display=\"inline\"><mi>f<\/mi><\/math> und wird auch mit <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> oder <math display=\"inline\"><mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/math> bezeichnet. Falls die unabh\u00e4ngige Variable <math display=\"inline\"><mi>t<\/mi><\/math> ist, schreiben wir <math display=\"inline\"><mover accent=\"true\"><mrow><mi>f<\/mi><\/mrow><mo accent=\"true\">\u00a8<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u1e1f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u22c5<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/math> f\u00fcr die zweite Ableitung nach <span class=\"maperiod\"><math display=\"inline\"><mi>t<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere erhalten wir, dass eine zweimal differenzierbare Funktion <math display=\"inline\"><mi>f<\/mi><\/math> stetig differenzierbar ist. <\/p><p class=\"indent\">Induktiv kann man nun h\u00f6here Differenzierbarkeit und h\u00f6here Ableitungen definieren. Formal definieren wir also die Ableitungen <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>2<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mspace class=\"quad\" width=\"1em\" \/><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><msup><mrow><mi class=\"qopname\">d<\/mi><mo>  <\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mi>f<\/mi><\/mrow> <mrow><mspace class=\"thinspace\" width=\"0.17em\" \/><mi class=\"qopname\">d<\/mi><mo>  <\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> Falls <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <\/math> f\u00fcr ein <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> (auf ganz <math display=\"inline\"><mi>D<\/mi><\/math>) existiert, heisst <math display=\"inline\"><mi>f<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecbx-1095\">-mal differenzierbar<\/span>. Falls die <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecbx-1095\">-te<\/span> <span class=\"ecbx-1095\">Ableitung <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> zus\u00e4tzlich stetig ist, heisst <math display=\"inline\"><mi>f<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecbx-1095\">-mal stetig differenzierbar<\/span>. Die Menge der <math display=\"inline\"><mi>n<\/mi><\/math>-mal stetig differenzierbaren Funktionen auf <math display=\"inline\"><mi>D<\/mi><\/math> bezeichnen wir mit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>C<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span><a id=\"dx1-231001\"><\/a> <\/p><p class=\"indent\">F\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> kann man eine Funktion finden, die zwar <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbar, aber nicht <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math>-mal differenzierbar ist. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1c58f798483a\"> <a id=\"x1-231002r21\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.21.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Zeigen Sie, dass die Funktion <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><mo class=\"MathClass-rel\">|<\/mo><mi>x<\/mi><mo class=\"MathClass-rel\">|<\/mo><mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">stetig differenzierbar, aber nicht <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbar ist.<\/span> <\/p> <\/div> <p class=\"indent\">Wir sagen, dass <math display=\"inline\"><mi>f<\/mi><\/math> <span class=\"ecbx-1095\">glatt <\/span>oder <span class=\"ecbx-1095\">beliebig oft differenzierbar <\/span>ist, falls <math display=\"inline\"><mi>f<\/mi><\/math> f\u00fcr jedes <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbar ist. Ist <math display=\"inline\"><mi>f<\/mi><\/math> glatt, so sind insbesondere alle Ableitungen von <math display=\"inline\"><mi>f<\/mi><\/math> stetig (<math display=\"inline\"><mi>f<\/mi><\/math> ist also beliebig oft stetig differenzierbar). Die Menge der glatten Funktionen auf <math display=\"inline\"><mi>D<\/mi><\/math> bezeichnen wir mit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>C<\/mi><\/mrow><mrow><mi>\u221e<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mi>D<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\">Wir kennen bereits einige Beispiele glatter Funktionen. Dazu geh\u00f6ren die Polynome, da diese nach Korollar <a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\">8.6<\/a> differenzierbar sind und da deren Ableitung ein Polynom ist, womit die Aussage aus Induktion folgt. Ebenfalls glatt sind die Funktion <math display=\"inline\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> sin<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\"> cos<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">sinh<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">,<\/mo><mi class=\"qopname\">cosh<\/mi><mo>  <\/mo><\/math> nach Beispiel <a href=\"..\/..\/chapter\/die-ableitung#x1-228001r3\">8.3<\/a> und \u00dcbung <a href=\"..\/..\/chapter\/die-ableitung#x1-228005r4\">8.4<\/a>. Etwas interessanter, aber nicht ganz unerwartet ist vermutlich folgendes Beispiel. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z8fb880fdd535\"> <a id=\"x1-231003r22\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.22 <\/span>(Logarithmusfunktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Der Logarithmus <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo> <mo class=\"MathClass-punc\">:<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-punc\">,<\/mo><mi>\u221e<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo><mi class=\"qopname\"> log<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">ist glatt. In der Tat gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2033<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac> <mrow> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mn>3<\/mn><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mfrac> <mrow> <mn>2<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">oder allgemein <\/span><span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>n<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">!<\/mo><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">was sich mit vollst<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ndiger Induktion beweisen l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst.<\/span> <\/p> <\/div> <p class=\"indent\">Ein \u00fcberraschenderes Beispiel einer glatten Funktion ist vielleicht das folgende. <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zcdd1483c9845\"> <a id=\"x1-231004r23\"><\/a> <span class=\"ecbx-1095\">Beispiel 8.23 <\/span>(Glattes Abklingen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die Funktion <\/span><math display=\"inline\"><mi>\u03c8<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">definiert durch<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow> <mtable align=\"axis\" class=\"array\" columnlines=\"none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"> <mn>0<\/mn> <\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2264<\/mo> <mn>0<\/mn><\/mtd> <\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd><mtd class=\"array\" columnalign=\"left\"><mstyle class=\"text\"><mtext>falls&nbsp;<\/mtext><\/mstyle><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/mtd><\/mtr> <\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">ist glatt und demnach auch beliebig oft stetig differenzierbar, siehe das folgende Bild.<\/span> <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=566&amp;url=Pictures\/ableitung\/expm1overx.pdf\"><img decoding=\"async\" id=\"ze7309b3f1788\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/ableitung\/expm1overx.svg\" width=\"566\" \/><\/div>  <\/div> <p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gibt es nichts zu zeigen, da die Ableitung der Nullfunktion die Nullfunktion ist. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">ergibt sich dies mittels Induktion, der Kettenregel (Satz <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228014r8\"><span class=\"ecti-1095\">8.8<\/span><\/a><span class=\"ecti-1095\">), Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228015r9\"><span class=\"ecti-1095\">8.9<\/span><\/a><span class=\"ecti-1095\">,<\/span> <span class=\"ecti-1095\">der Produktregel in Proposition <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\"><span class=\"ecti-1095\">8.5<\/span><\/a> <span class=\"ecti-1095\">und Korollar <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-228011r6\"><span class=\"ecti-1095\">8.6<\/span><\/a><span class=\"ecti-1095\">. In der Tat gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mi class=\"qopname\">\u2033<\/mi><mo>  <\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">und (da die konkrete Formel f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">schnell kompliziert wird) allgemeiner<\/span> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-231005r4\" \/><mstyle class=\"maketag\"><mtext>(8.4)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r gewisse Polynome <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">und jedes <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>2<\/mn><\/math> <span class=\"ecti-1095\">haben wir diese Darstellung der Ableitung bereits bewiesen, wobei<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">und<\/span> <math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>4<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>2<\/mn><msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msup><\/math><span class=\"ecti-1095\">. F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Induktionsschritt<\/span> <span class=\"ecti-1095\">nehmen wir<\/span> (<a href=\"..\/..\/chapter\/die-ableitung#x1-231005r4\">8.4<\/a>) <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">an und erhalten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi class=\"qopname\">exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">wobei das Polynom <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msub><\/math> <span class=\"ecti-1095\">als <\/span><math display=\"inline\"><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>t<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msubsup><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msubsup><mo class=\"MathClass-open\">(<\/mo><mi>t<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">gew<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">hlt wurde.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Es bleibt noch zu zeigen, dass <\/span><math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">auch in <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">beliebig oft differenzierbar ist. Dabei k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnen wir nicht auf unsere Ableitungsregeln zur<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ckgreifen,<\/span> <span class=\"ecti-1095\">sondern m<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">ssen dies direkt mit der Definition der Ableitung <\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">berpr<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">fen. Wir behaupten, dass<\/span> <math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo> <\/mrow> <\/msup> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r den Beweis der Behauptung zeigen wir zuerst, dass f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r jedes Polynom<\/span> <math display=\"inline\"><mi>f<\/mi><\/math> <\/p><math display=\"block\"><mtable class=\"align\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi>f<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mfrac><mrow> <mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"><mstyle class=\"label\" id=\"x1-231006r5\" \/><mstyle class=\"maketag\"><mtext>(8.5)<\/mtext><\/mstyle><mspace class=\"nbsp\" width=\"0.33em\" \/> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">ist. Auf Grund der Linearit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">t des Grenzwerts und da<\/span> <math display=\"inline\"><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">gilt, gen<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">gt es<\/span> <span class=\"ecti-1095\">zu zeigen, dass <\/span><math display=\"inline\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2198<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>n<\/mi><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">gilt.<\/span> <span class=\"ecti-1095\">Setzen wir <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so erhalten wir, dass diese Behauptung wiederum zu<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><munder class=\"msub\"><mrow><mi class=\"qopname\">lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>y<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mi>\u221e<\/mi><\/mrow><\/munder> <mfrac><mrow><msup><mrow><mi>y<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msup><\/mrow> <mrow><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">quivalent ist. Dies folgt aber mit dem Sandwich-Lemma aus der Ungleichung<\/span> <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mfrac> <mrow> <mi>y<\/mi><\/mrow> <mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">\u2264<\/mo><mi class=\"qopname\"> exp<\/mi><mo>  <\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <span class=\"ecti-1095\">alle<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">und<\/span><span class=\"ecti-1095\">&nbsp;<\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">(siehe Abschnitt <\/span><a href=\"..\/..\/chapter\/die-exponentialfunktion#x1-1650003\"><span class=\"ecti-1095\">6.3<\/span><\/a><span class=\"ecti-1095\">).<\/span> <\/p><p class=\"indent\"><span class=\"ecti-1095\">Wir zeigen nun <\/span><math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">per Induktion. Verwenden wir<\/span> (<a href=\"..\/..\/chapter\/die-ableitung#x1-231006r5\">8.5<\/a>)<span class=\"ecti-1095\">, so erhalten wir<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Falls wir bereits <\/span><math display=\"inline\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r ein <\/span><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> <span class=\"ecti-1095\">wissen, dann folgt ebenso<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">+<\/mo><mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mn>0<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow> <mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <msup><mrow><mi>\u03c8<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mfrac><mrow><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>0<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo><munder class=\"msub\"><mrow><mi class=\"qopname\"> lim<\/mi><mo>  <\/mo><\/mrow><mrow><mi>x<\/mi><mo class=\"MathClass-rel\">\u2192<\/mo><mn>0<\/mn><\/mrow><\/munder><mi>\u03c8<\/mi> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><msub><mrow><mi>f<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac><mstyle><mrow><mo fence=\"true\" form=\"prefix\"> )<\/mo><mrow \/><mo fence=\"true\" form=\"postfix\" \/><\/mrow><\/mstyle><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>x<\/mi><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Wir haben nun also gezeigt, dass alle Ableitungen von<\/span> <math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">auf ganz<\/span> <math display=\"inline\"><mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">existieren und<\/span> <span class=\"ecti-1095\">somit ist <\/span><math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">glatt.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze34dbb788ba5\"> <a id=\"x1-231007r24\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.24 <\/span>(Hutfunktion)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Finden Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r beliebige reelle Zahlen <\/span><math display=\"inline\"><mi>a<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>b<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>c<\/mi> <mo class=\"MathClass-rel\">&lt;<\/mo> <mi>d<\/mi><\/math> <span class=\"ecti-1095\">eine glatte Funktion <\/span><math display=\"inline\"><mi>\u03c6<\/mi><\/math> <span class=\"ecti-1095\">auf <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">so dass <\/span><math display=\"inline\"><mi>\u03c6<\/mi><\/math> <span class=\"ecti-1095\">gleich Null ist ausserhalb des Intervalls <\/span><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">und gleich <\/span><math display=\"inline\"><mn>1<\/mn><\/math> <span class=\"ecti-1095\">ist auf dem Intervall <\/span><span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">[<\/mo><mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>c<\/mi><mo class=\"MathClass-close\">]<\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Versuchen Sie zuerst geeignet verschobene und gespiegelte Versionen der Funktion<\/span> <math display=\"inline\"><mi>\u03c8<\/mi><\/math> <span class=\"ecti-1095\">aus Beispiel <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-231004r23\"><span class=\"ecti-1095\">8.23<\/span><\/a> <span class=\"ecti-1095\">zu  kombinieren.  Zum  Start  k<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">nnte  man  beispielsweise  die  Abbildung<\/span> <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo> <mspace class=\"nbsp\" width=\"0.33em\" \/> <mfrac> <mrow> <mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow> <mrow><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">+<\/mo><mi>\u03c8<\/mi><mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/mfrac><\/math> <span class=\"ecti-1095\">betrachten.<\/span><\/p><\/details>  <\/div> <p class=\"indent\">Wir wenden uns nun wieder allgemeinen Aussagen im Stile von Abschnitt <a href=\"..\/..\/chapter\/die-ableitung#x1-2280002\">8.1.2<\/a> zu. Aus Proposition <a href=\"..\/..\/chapter\/die-ableitung#x1-228010r5\">8.5<\/a> l\u00e4sst sich folgendes Korollar deduzieren. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze6e556d944e3\"> <a id=\"x1-231008r25\"><\/a> <span class=\"ecbx-1095\">Korollar 8.25 <\/span>(Summen und Produkte bei h\u00f6herer Differenzierbarkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Sei <\/span><math display=\"inline\"><mi>D<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine Teilmenge,<\/span> <span class=\"ecti-1095\">so dass jeder Punkt in <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">ist. Seien <\/span><math display=\"inline\"><mi>f<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal differenzierbar.<\/span> <span class=\"ecti-1095\">Dann sind <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>g<\/mi><\/math> <span class=\"ecti-1095\">ebenso<\/span> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal differenzierbar<\/span> <span class=\"ecti-1095\">und es gilt <\/span><math display=\"inline\"><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>g<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">sowie<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mi>g<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><munderover accent=\"false\" accentunder=\"false\"><mrow><mo> \u2211<\/mo> <\/mrow><mrow><mi>k<\/mi><mo class=\"MathClass-rel\">=<\/mo><mn>0<\/mn><\/mrow><mrow><mi>n<\/mi><\/mrow><\/munderover><mfenced close=\")\" open=\"(\" separators><mfrac linethickness=\"0.0pt\"><mrow><mi>n<\/mi><\/mrow> <mrow><mi>k<\/mi><\/mrow><\/mfrac><\/mfenced><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-bin\">\u2212<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">Insbesondere ist jedes skalare Vielfache <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbar und <\/span><math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>\u03b1<\/mi><mi>f<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mi>\u03b1<\/mi><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u03b1<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">Die obige Produktregel f\u00fcr h\u00f6here Ableitungen nennt sich auch <span class=\"ecbx-1095\">Leibniz-Regel<\/span>.                                                                                                                                                                           <\/p><p class=\"indent\">Nat\u00fcrlich sind auch Verkn\u00fcpfungen von <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbaren Funktionen <math display=\"inline\"><mi>n<\/mi><\/math>-mal differenzierbar. Allerdings ist es im Gegensatz zum Produkt deutlich schwerer, hier eine explizite Formel anzugeben. Wir beschr\u00e4nken uns deswegen darauf, nur die Differenzierbarkeit zu formulieren. <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z194ade77368a\"> <a id=\"x1-231009r26\"><\/a> <span class=\"ecbx-1095\">Korollar 8.26 <\/span>(Verkn\u00fcpfungen und h\u00f6here Differenzierbarkeit)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Seien <\/span><math display=\"inline\"><mi>D<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>E<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">Teilmengen, so dass jeder Punkt in <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><mi>E<\/mi><\/math> <span class=\"ecti-1095\">ein H<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ufungspunkt von <\/span><math display=\"inline\"><mi>D<\/mi><\/math> <span class=\"ecti-1095\">respektive <\/span><math display=\"inline\"><mi>E<\/mi><\/math> <span class=\"ecti-1095\">ist. Sei des Weiteren <\/span><math display=\"inline\"><mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>E<\/mi><\/math> <span class=\"ecti-1095\">eine <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbare Funktion und sei <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>E<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">eine <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbare Funktion. Dann ist <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi> <mo class=\"MathClass-punc\">:<\/mo> <mi>D<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u211d<\/mi><\/math> <math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-mal<\/span> <span class=\"ecti-1095\">differenzierbar.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z338913fa3c12\"> <a id=\"x1-231010r27\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 8.27.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Beweisen Sie die Korollare <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-231008r25\"><span class=\"ecti-1095\">8.25<\/span><\/a> <span class=\"ecti-1095\">und <\/span><a href=\"..\/..\/chapter\/die-ableitung#x1-231009r26\"><span class=\"ecti-1095\">8.26<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p><div class=\"wp-nocaption \"><\/div><details><summary style=\"color:#FF7F00\"><span class=\"ecti-1095\">Hinweis.<\/span><\/summary><p class=\"indent\" style=\"margin-top: 0\"> <span class=\"ecti-1095\">Die <\/span><math display=\"inline\"><mi>n<\/mi><\/math><span class=\"ecti-1095\">-te<\/span> <span class=\"ecti-1095\">Ableitung von <\/span><math display=\"inline\"><mi>g<\/mi> <mo class=\"MathClass-bin\">\u2218<\/mo> <mi>f<\/mi><\/math> <span class=\"ecti-1095\">ist eine Linearkombination von Funktionen der Form <\/span><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>D<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><msup><mrow><mi>g<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>k<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>f<\/mi><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo>\u2032<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><msub><mrow><mi>k<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u22ef<\/mo><msup><mrow><mo class=\"MathClass-open\">(<\/mo><msup><mrow><mi>f<\/mi><\/mrow><mrow><mo class=\"MathClass-open\">(<\/mo><mi>n<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mrow><\/msup><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><msub><mrow><mi>k<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub><\/mrow><\/msup><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2265<\/mo> <mi>k<\/mi><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>k<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><mi class=\"MathClass-op\">\u2026<\/mi><mo> <\/mo><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>k<\/mi><\/mrow><mrow><mi>n<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span><\/p><\/details>  <\/div> <a id=\"x1-231011r226\"><\/a> 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