对数微分法

对数微分法（英語：Logarithmic differentiation）是在微积分学中，通过求某函数f的对数导数（英语：Logarithmic derivative）来求得函数导数的一种方法， [ ln ⁡ ( f ) ] ′ = f ′ f → f ′ = f ⋅ [ ln ⁡ ( f ) ] ′ . {\displaystyle [\ln(f)]'={\frac {f'}{f}}\quad \rightarrow \quad f'=f\cdot [\ln(f)]'.} 这一方法常在函数对数求导比对函数本身求导更容易时使用，这样的函数通常是几项的积，取对数之后，可以把函数变成容易求导的几项的和。这一方法对幂函数形式的函数也很有用。对数微分法依赖于链式法则和对数的性质（尤其是自然对数），把积变为求和，把商变为做差。这一方法可以应用于所有恆不为0的可微函数。 对于某函数 y = f ( x ) {\displaystyle y=f(x)\,\!} 运用对数微分法，通常对函数两边取绝对值后取自然对数。 ln ⁡ | y | = ln ⁡ | f ( x ) | {\displaystyle \ln |y|=\ln |f(x)|\,\!} 运用隐式微分法，可得 1 y d y d x = f ′ ( x ) f ( x ) {\displaystyle {\frac {1}{y}}{\frac {dy}{dx}}={\frac {f'(x)}{f(x)}}} 两边同乘以y，则方程左边只剩下dy/dx： d y d x = y × f ′ ( x ) f ( x ) = f ′ ( x ) . {\displaystyle {\frac {dy}{dx}}=y\times {\frac {f'(x)}{f(x)}}=f'(x).} 对数微分法有用，是因为对数的性质可以大大简化复杂函数的微分，常用的对数性质有： ln ⁡ ( a b ) = ln ⁡ ( a ) + ln ⁡ ( b ) , ln ⁡ ( a b ) = ln ⁡ ( a ) − ln ⁡ ( b ) , ln ⁡ ( a n ) = n ln ⁡ ( a ) {\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)} 有一如下形式的函数， f ( x ) = ∏ i ( f i ( x ) ) α i ( x ) . {\displaystyle f(x)=\prod _{i}(f_{i}(x))^{\alpha _{i}(x)}.} 两边取自然对数，得 ln ⁡ ( f ( x ) ) = ∑ i α i ( x ) ⋅ ln ⁡ ( f i ( x ) ) , {\displaystyle \ln(f(x))=\sum _{i}\alpha _{i}(x)\cdot \ln(f_{i}(x)),} 两边对x求导，得 f ′ ( x ) f ( x ) = ∑ i [ α i ′ ( x ) ⋅ ln ⁡ ( f i ( x ) ) + α i ( x ) ⋅ f i ′ ( x ) 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].} 两边同乘以 f ( x ) {\displaystyle f(x)} ，可得原函数的导数为 f ′ ( x ) = ∏ i ( f i ( x ) ) α i ( x ) ⏞ f ( x ) × ∑ i { α i ′ ( x ) ⋅ ln ⁡ ( f i ( x ) ) + α i ( x ) ⋅ f i ′ ( x ) f i ( x ) } ⏞ [ ln ⁡ ( f ( x ) ) ] ′ {\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))]'}} 对如下形式的两个函数的积函数 f ( x ) = g ( x ) h ( x ) {\displaystyle f(x)=g(x)h(x)\,\!} 两边取自然对数，可得如下形式的和函数 ln ⁡ ( f ( x ) ) = ln ⁡ ( g ( x ) h ( x ) ) = ln ⁡ ( g ( x ) ) + ln ⁡ ( h ( x ) ) {\displaystyle \ln(f(x))=\ln(g(x)h(x))=\ln(g(x))+\ln(h(x))\,\!} 应用链式法则，两边微分，得 f ′ ( x ) f ( x ) = g ′ ( x ) g ( x ) + h ′ ( x ) h ( x ) {\displaystyle {\frac {f'(x)}{f(x)}}={\frac {g'(x)}{g(x)}}+{\frac {h'(x)}{h(x)}}} 整理，可得 f ′ ( x ) = f ( x ) × { g ′ ( x ) g ( x ) + h ′ ( x ) h ( x ) } = g ( x ) h ( x ) × { g ′ ( x ) g ( x ) + 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 \}}} 对如下形式的两个函数的商函数 f ( x ) = g ( x ) h ( x ) {\displaystyle f(x)={\frac {g(x)}{h(x)}}\,\!} 两边取自然对数，可得如下形式的差函数 ln ⁡ ( f ( x ) ) = ln ⁡ ( g ( x ) h ( x ) ) = ln ⁡ ( g ( x ) ) − ln ⁡ ( h ( x ) ) {\displaystyle \ln(f(x))=\ln {\Bigg (}{\frac {g(x)}{h(x)}}{\Bigg )}=\ln(g(x))-\ln(h(x))\,\!} 应用链式法则，两边求导，得 f ′ ( x ) f ( x ) = g ′ ( x ) g ( x ) − h ′ ( x ) h ( x ) {\displaystyle {\frac {f'(x)}{f(x)}}={\frac {g'(x)}{g(x)}}-{\frac {h'(x)}{h(x)}}} 整理，可得 f ′ ( x ) = f ( x ) × { g ′ ( x ) g ( x ) − h ′ ( x ) h ( x ) } = g ( x ) h ( x ) × { g ′ ( x ) g ( x ) − 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 \}}} 右边通分之后，结果和对 f ( x ) {\displaystyle f(x)} 运用除法定则所得结果相同。 对于如下形式的函数 f ( x ) = g ( x ) h ( x ) {\displaystyle f(x)=g(x)^{h(x)}\,\!} 两边取自然对数，可得如下形式的积函数 ln ⁡ ( f ( x ) ) = ln ⁡ ( g ( x ) h ( x ) ) = h ( x ) ln ⁡ ( g ( x ) ) {\displaystyle \ln(f(x))=\ln \left(g(x)^{h(x)}\right)=h(x)\ln(g(x))\,\!} 应用链式法则，两边求导，得 f ′ ( x ) f ( x ) = h ′ ( x ) ln ⁡ ( g ( x ) ) + h ( x ) g ′ ( x ) g ( x ) {\displaystyle {\frac {f'(x)}{f(x)}}=h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)}}} 整理，得 f ′ ( x ) = f ( x ) × { h ′ ( x ) ln ⁡ ( g ( x ) ) + h ( x ) g ′ ( x ) g ( x ) } = g ( x ) h ( x ) × { h ′ ( x ) ln ⁡ ( g ( x ) ) + h ( x ) 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 \}}.} 与将函数f看做指数函数，直接运用链式法则所得结果相同。 数学主题 对数恒等式 Krantz, Steven G. 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