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<title>Logarithmische Ableitung</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Logarithmische Ableitung</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">In der <a href="Analysis" title="Analysis">Analysis</a> ist die logarithmische Ableitung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {L} (f)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a000f6d03abdc0a8a68b54e76cbab6993123f213.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \operatorname {L} (f)}" loading="lazy"> einer differenzierbaren <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, die keine <a href="Nullstelle" title="Nullstelle">Nullstellen</a> besitzt, als der Quotient der <a href="Differentialrechnung" title="Differentialrechnung">Ableitung</a> einer Funktion und der Funktion selbst definiert:<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (f):={\frac {f'}{f}}.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5315cbd7e9aa679374fe28d2cd7948286baa4ca8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.774ex; height:6.009ex;" alt="{\displaystyle \operatorname {L} (f):={\frac {f'}{f}}.}" loading="lazy"></dd></dl>

Auf gleiche Weise lässt sich der Begriff auch für von Null verschiedene <a href="Meromorphe_Funktion" title="Meromorphe Funktion">meromorphe Funktionen</a> definieren (hier brauchen keine Nullstellen ausgeschlossen zu werden, weil der Quotient für meromorphe Funktionen <a href="Wohldefiniertheit" title="Wohldefiniertheit">wohldefiniert</a> ist).

Für reelle Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> mit positiven Werten stimmt die logarithmische Ableitung nach der <a href="Kettenregel" title="Kettenregel">Kettenregel</a> mit der Ableitung der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln(f)}">

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<annotation encoding="application/x-tex">{\displaystyle \ln(f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ab8a5198fcc93dcf6cff138e74820dd39428ed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.027ex; height:2.843ex;" alt="{\displaystyle \ln(f)}" loading="lazy"> überein; daher der Name. Es gilt also

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<annotation encoding="application/x-tex">{\displaystyle (\ln f)'={\frac {f'}{f}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/31061605d479bd257298b2255c714ee547e8e490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.039ex; height:6.009ex;" alt="{\displaystyle (\ln f)'={\frac {f'}{f}}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Rechenregeln">Rechenregeln</h2></div>

Die Bedeutung des Begriffes liegt in der Formel für die logarithmische Ableitung eines Produktes:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (f\cdot g)=\operatorname {L} (f)+\operatorname {L} (g)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03a6bca64ebe60424bb46be269137fbf72086954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.193ex; height:2.843ex;" alt="{\displaystyle \operatorname {L} (f\cdot g)=\operatorname {L} (f)+\operatorname {L} (g)}" loading="lazy">,</dd></dl>

allgemein

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (f_{1}\cdots f_{n})=\operatorname {L} (f_{1})+\ldots +\operatorname {L} (f_{n})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f0fc2dfd8beef0a5e65e74e67f3c84e833bc3976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.888ex; height:2.843ex;" alt="{\displaystyle \operatorname {L} (f_{1}\cdots f_{n})=\operatorname {L} (f_{1})+\ldots +\operatorname {L} (f_{n})}" loading="lazy">.</dd></dl>

Als Abwandlung zur Produktregel gilt also

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<annotation encoding="application/x-tex">{\displaystyle (fg)'=fg(\operatorname {L} (f)+\operatorname {L} (g))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2570a595ad2276005d332d36b6f3dcbcfdf25c5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.95ex; height:3.009ex;" alt="{\displaystyle (fg)'=fg(\operatorname {L} (f)+\operatorname {L} (g))}" loading="lazy">.</dd></dl>

Analog gilt

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (1/f)=-\operatorname {L} (f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad9e52a7e1a9e2bafe20e4d408e92fe4203031ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.7ex; height:2.843ex;" alt="{\displaystyle \operatorname {L} (1/f)=-\operatorname {L} (f)}" loading="lazy"></dd></dl>

und

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (f/g)=\operatorname {L} (f)-\operatorname {L} (g)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/575d6cfdd838a895b258b89f0418c8261120b64a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.677ex; height:2.843ex;" alt="{\displaystyle \operatorname {L} (f/g)=\operatorname {L} (f)-\operatorname {L} (g)}" loading="lazy">.</dd></dl>

Für die logarithmische Ableitung der <a href="Potenzfunktion" title="Potenzfunktion">Potenzfunktion</a> erhält man etwa

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (f^{n})=n\cdot \operatorname {L} (f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81f3bce9f645abe295467aeddeda1a20c7ea47b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.514ex; height:2.843ex;" alt="{\displaystyle \operatorname {L} (f^{n})=n\cdot \operatorname {L} (f)}" loading="lazy">.</dd></dl>

Diese Formeln folgen aus der <a href="Produktregel" title="Produktregel">Leibnizregel</a> und gelten deshalb auch in allgemeinerem Kontext, beispielsweise bei der (formalen) Ableitung von <a href="Polynom" title="Polynom">Polynomen</a> oder <a href="Rationale_Funktion" title="Rationale Funktion">rationalen Funktionen</a> über einem beliebigen Grund<a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">körper</a>.

<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>

Die logarithmische Ableitung von Funktionen kann meistens mit den normalen <a href="Tabelle_von_Ableitungs-_und_Stammfunktionen" title="Tabelle von Ableitungs- und Stammfunktionen">Differentiationsregeln</a> bestimmt werden.

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></th>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {L} (f)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a000f6d03abdc0a8a68b54e76cbab6993123f213.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.541ex; height:2.843ex;" alt="{\displaystyle \operatorname {L} (f)}" loading="lazy"></th>

<th>Anmerkungen

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {R} \setminus \{0\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe49cb3bf2897ac960370aa296d8a5640d5c7ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.596ex; height:2.843ex;" alt="{\displaystyle n\in \mathbb {R} \setminus \{0\}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle x^{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/150d38e238991bc4d0689ffc9d2a852547d2658d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.548ex; height:2.343ex;" alt="{\displaystyle x^{n}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {n}{x}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3017bff0f0ef95c73cf57d969c3cf7afb347f98f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.231ex; height:4.676ex;" alt="{\displaystyle {\frac {n}{x}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f631abf6775cd6861c55305e8e0f6845df6daf11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {R} }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle e^{nx}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9c7e16dfc1eaddd7d60dc04d98f89c9a55b20ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.242ex; height:2.343ex;" alt="{\displaystyle e^{nx}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {R} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f631abf6775cd6861c55305e8e0f6845df6daf11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {R} }" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \ln(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0df055b8e294310e6785701c1c67105e109191d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.078ex; height:2.843ex;" alt="{\displaystyle \ln(x)}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{x\ln(x)}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f1ed4ebe32db0f8580ac87c537e416b1fbdd8d6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:7.631ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{x\ln(x)}}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \sin(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a990a5545cac26c1c6821dca95d898bc80fe3a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.995ex; height:2.843ex;" alt="{\displaystyle \sin(x)}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \cot(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a94de2d0fad8847a668fd46eaae836b95b33613.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.238ex; height:2.843ex;" alt="{\displaystyle \cot(x)}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \cos(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb9af7ed6f44822021b74bb82b431022c7fd66b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.25ex; height:2.843ex;" alt="{\displaystyle \cos(x)}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle -\tan(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3decf658510d27e760866e328941f605eb73d389.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.694ex; height:2.843ex;" alt="{\displaystyle -\tan(x)}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \tan(x)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/398da5ded10e1ab022cfc8c3f4a4a87b46cd8c46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.499ex; height:2.843ex;" alt="{\displaystyle \tan(x)}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sin(x)\cos(x)}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff5bd7f6f2e80accb0269ec9f3df945c7342eb95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.468ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{\sin(x)\cos(x)}}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11ca17f880240539116aac7e6326909299e2a080.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.35ex; height:2.843ex;" alt="{\displaystyle \Gamma (z)}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \psi (z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c02965f8dd8bfe2c0352b07c1193b8dc276c1d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.411ex; height:2.843ex;" alt="{\displaystyle \psi (z)}" loading="lazy">

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<td style="text-align:left">Die logarithmische Ableitung der <a href="Gamma-Funktion" class="mw-redirect" title="Gamma-Funktion">Gamma-Funktion</a> ist die <a href="Digamma-Funktion" title="Digamma-Funktion">Digamma-Funktion</a>.

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<div class="mw-heading mw-heading2"><h2 id="Funktionentheorie">Funktionentheorie</h2></div>

Es sei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(z)}">

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<annotation encoding="application/x-tex">{\displaystyle g(z)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8eb6fb9c0f0d402f5ebbebfd5e34bedd39a4a52b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.013ex; height:2.843ex;" alt="{\displaystyle g(z)}" loading="lazy"> eine <a href="Meromorphe_Funktion" title="Meromorphe Funktion">meromorphe Funktion</a> mit einer Nullstelle der Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> oder einem Pol der Ordnung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -n}">

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<annotation encoding="application/x-tex">{\displaystyle -n}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f00139753ecf4fe00a10a17bd5620b70a61b29e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle -n}" loading="lazy"> an einer Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9e8fc86339df069b29043cc911848d044ec3f670.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.526ex; height:2.176ex;" alt="{\displaystyle c\in \mathbb {C} }" loading="lazy">. Dann lässt sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8eb6fb9c0f0d402f5ebbebfd5e34bedd39a4a52b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.013ex; height:2.843ex;" alt="{\displaystyle g(z)}" loading="lazy"> als

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(z)=(z-c)^{n}f(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3bf18308fc7b7a94f8820143a89a5876aac8bdc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.251ex; height:2.843ex;" alt="{\displaystyle g(z)=(z-c)^{n}f(z)}" loading="lazy"></dd></dl>

mit einer in einer Umgebung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> holomorphen Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(z)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d8dd568d570b390c337c0a911f0a1c5c214e8240.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.176ex; height:2.843ex;" alt="{\displaystyle f(z)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c)\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c)\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1791edfa992c2013a71c4372f869be2f8c9a2c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.356ex; height:2.843ex;" alt="{\displaystyle f(c)\neq 0}" loading="lazy"> schreiben. Es gilt

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(g)={\frac {n}{z-c}}+L(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2ab501eb8df0c56326ecddb494370c0aee05c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.889ex; height:4.843ex;" alt="{\displaystyle L(g)={\frac {n}{z-c}}+L(f)}" loading="lazy">.</dd></dl>

Wegen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c)\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(c)\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1791edfa992c2013a71c4372f869be2f8c9a2c93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.356ex; height:2.843ex;" alt="{\displaystyle f(c)\neq 0}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64d16ee522cba7990a6dbd9b202696553309a3f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.671ex; height:2.843ex;" alt="{\displaystyle L(f)}" loading="lazy"> in einer Umgebung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> holomorph. Das <a href="Residuum_(Funktionentheorie)" title="Residuum (Funktionentheorie)">Residuum</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(g)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(g)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b987f80984cf1f775f83301fd5319d3f11197fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.508ex; height:2.843ex;" alt="{\displaystyle L(g)}" loading="lazy"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> entspricht also gerade der Nullstellenordnung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> an der Stelle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">. Dieser Zusammenhang wird im <a href="Prinzip_vom_Argument" title="Prinzip vom Argument">Prinzip vom Argument</a> ausgenutzt.

<div class="mw-heading mw-heading2"><h2 id="Anwendung">Anwendung</h2></div>

Lässt sich eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> darstellen als

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=k\cdot u^{a}\cdot v^{b}\cdot w^{c}\cdot \dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/67b6e8838794257a26243023d82fa11eda06eab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.133ex; height:3.009ex;" alt="{\displaystyle f=k\cdot u^{a}\cdot v^{b}\cdot w^{c}\cdot \dots }" loading="lazy"></dd></dl>

mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,\dots }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a,b,c,\dots }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/948b9e9c25a46ba3515e07facbbee84dd8db5bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.059ex; height:2.509ex;" alt="{\displaystyle a,b,c,\dots }" loading="lazy"> als Konstanten, so ergibt sich die Ableitung zu

<msup>

</msup>

<mrow>

<mrow>

<mfrac>

<msup>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</msup>

</mfrac>

</mrow>

<mfrac>

<msup>

</msup>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'=f\cdot \left(a\cdot {\frac {u'}{u}}+b\cdot {\frac {v'}{v}}+c\cdot {\frac {w'}{w}}+\dots \right).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c21d82d7b6e5a87b2161260f7774c0af953490b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.716ex; height:6.176ex;" alt="{\displaystyle f'=f\cdot \left(a\cdot {\frac {u'}{u}}+b\cdot {\frac {v'}{v}}+c\cdot {\frac {w'}{w}}+\dots \right).}" loading="lazy"></dd></dl>

Dieser Umstand kann bei praktischen Anwendungen wie der Handrechnung genutzt werden, um manche <a href="Differentialrechnung#Ableitungsregeln" title="Differentialrechnung">Ableitungsregeln</a> kompakt zusammenzufassen: So ergibt sich beispielsweise bei den Faktoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6104442ed30596ef4d7795d3186273f68d796ea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f55bc77dec8088791b5c1ed51e634cc1b431fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.258ex; height:2.176ex;" alt="{\displaystyle b=1}" loading="lazy"> die <a href="Produktregel" title="Produktregel">Produktregel</a>, mit den Faktoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6104442ed30596ef4d7795d3186273f68d796ea4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b=-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fed9de3e9fd089ca51cac7892522579fdad649e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.067ex; height:2.343ex;" alt="{\displaystyle b=-1}" loading="lazy"> die <a href="Quotientenregel" title="Quotientenregel">Quotientenregel</a> und mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=-1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/231103d8099e125875dd690668e93a56aa10bd99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.299ex; height:2.343ex;" alt="{\displaystyle a=-1}" loading="lazy"> die <a href="Reziprokenregel" title="Reziprokenregel">Reziprokenregel</a>.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Richard P. Feynman, Michael A. Gottlieb, Ralph Leighton: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Feynman’s Tips on Physics: A Problem-Solving Supplement to the Feynman Lectures on Physics.</cite> Addison-Wesley, San Francisco 2006, ISBN 0-8053-9063-4, Kapitel 1–4 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&rfr_id=info:sid/de.wikipedia.org:Logarithmische+Ableitung&rft.atitle=Kapitel+1-4&rft.au=Richard+P.+Feynman%2C+Michael+A.+Gottlieb%2C+Ralph+Leighton&rft.btitle=Feynman%E2%80%99s+Tips+on+Physics%3A+A+Problem-Solving+Supplement+to+the+Feynman+Lectures+on+Physics.&rft.date=2006&rft.genre=bookitem&rft.isbn=0805390634&rft.place=San+Francisco&rft.pub=Addison-Wesley" style="display:none"> </li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Konrad_K%C3%B6nigsberger" title="Konrad Königsberger">Konrad Königsberger</a>: <cite style="font-style:italic">Analysis 1</cite>. 6. Auflage. Springer, Berlin 2004, ISBN 3-540-40371-X, S. 143.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Logarithmische+Ableitung&rft.au=Konrad+K%C3%B6nigsberger&rft.btitle=Analysis+1&rft.date=2004&rft.edition=6.&rft.genre=book&rft.isbn=354040371X&rft.pages=143&rft.place=Berlin&rft.pub=Springer" style="display:none">

</li>

</ol></div><div><div class="zim-footer">

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