# 对数微分法

${\displaystyle [\ln(f)]'={\frac {f'}{f}}\quad \rightarrow \quad f'=f\cdot [\ln(f)]'.}$

## 概述

${\displaystyle y=f(x)\,\!}$

${\displaystyle \ln |y|=\ln |f(x)|\,\!}$

${\displaystyle {\frac {1}{y}}{\frac {dy}{dx}}={\frac {f'(x)}{f(x)}}}$

${\displaystyle {\frac {dy}{dx}}=y\times {\frac {f'(x)}{f(x)}}=f'(x).}$

${\displaystyle \ln(ab)=\ln(a)+\ln(b),\qquad \ln \left({\frac {a}{b}}\right)=\ln(a)-\ln(b),\qquad \ln(a^{n})=n\ln(a)}$

### 通用公式

${\displaystyle f(x)=\prod _{i}(f_{i}(x))^{\alpha _{i}(x)}.}$

${\displaystyle \ln(f(x))=\sum _{i}\alpha _{i}(x)\cdot \ln(f_{i}(x)),}$

${\displaystyle {\frac {f'(x)}{f(x)}}=\sum _{i}\left[\alpha _{i}'(x)\cdot \ln(f_{i}(x))+\alpha _{i}(x)\cdot {\frac {f_{i}'(x)}{f_{i}(x)}}\right].}$

${\displaystyle f'(x)=\overbrace {\prod _{i}(f_{i}(x))^{\alpha _{i}(x)}} ^{f(x)}\times \overbrace {\sum _{i}\left\{\alpha _{i}'(x)\cdot \ln(f_{i}(x))+\alpha _{i}(x)\cdot {\frac {f_{i}'(x)}{f_{i}(x)}}\right\}} ^{[\ln(f(x))]'}}$

## 应用

### 积函数

${\displaystyle f(x)=g(x)h(x)\,\!}$

${\displaystyle \ln(f(x))=\ln(g(x)h(x))=\ln(g(x))+\ln(h(x))\,\!}$

${\displaystyle {\frac {f'(x)}{f(x)}}={\frac {g'(x)}{g(x)}}+{\frac {h'(x)}{h(x)}}}$

${\displaystyle f'(x)=f(x)\times {\Bigg \{}{\frac {g'(x)}{g(x)}}+{\frac {h'(x)}{h(x)}}{\Bigg \}}=g(x)h(x)\times {\Bigg \{}{\frac {g'(x)}{g(x)}}+{\frac {h'(x)}{h(x)}}{\Bigg \}}}$

### 商函数

${\displaystyle f(x)={\frac {g(x)}{h(x)}}\,\!}$

${\displaystyle \ln(f(x))=\ln {\Bigg (}{\frac {g(x)}{h(x)}}{\Bigg )}=\ln(g(x))-\ln(h(x))\,\!}$

${\displaystyle {\frac {f'(x)}{f(x)}}={\frac {g'(x)}{g(x)}}-{\frac {h'(x)}{h(x)}}}$

${\displaystyle f'(x)=f(x)\times {\Bigg \{}{\frac {g'(x)}{g(x)}}-{\frac {h'(x)}{h(x)}}{\Bigg \}}={\frac {g(x)}{h(x)}}\times {\Bigg \{}{\frac {g'(x)}{g(x)}}-{\frac {h'(x)}{h(x)}}{\Bigg \}}}$

### 复合指数函数

${\displaystyle f(x)=g(x)^{h(x)}\,\!}$

${\displaystyle \ln(f(x))=\ln \left(g(x)^{h(x)}\right)=h(x)\ln(g(x))\,\!}$

${\displaystyle {\frac {f'(x)}{f(x)}}=h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)}}}$

${\displaystyle f'(x)=f(x)\times {\Bigg \{}h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)}}{\Bigg \}}=g(x)^{h(x)}\times {\Bigg \{}h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)}}{\Bigg \}}.}$