对数微分法 (英语:Logarithmic differentiation )是在微积分学 中,通过求某函数 f 的对数导数 来求得函数导数 的一种方法, [1]
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{\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[2] [3] 。这一方法可以应用于所有恒不为0的可微函数 。
概述
对于某函数
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{\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[4] 。
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{\displaystyle \ln|y| = \ln|f(x)|\,\!}
运用隐式微分法 [5] ,可得
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{\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 :
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{\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[6] ,常用的对数性质有:[3]
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{\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
有一如下形式的函数,
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{\displaystyle f(x)=\prod_i(f_i(x))^{\alpha_i(x)}.}
两边取自然对数,得
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{\displaystyle \ln (f(x))=\sum_i\alpha_i(x)\cdot \ln(f_i(x)),}
两边对x 求导,得
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{\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
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{\displaystyle f(x)}
,可得原函数的导数为
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{\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
对如下形式的两个函数的积函数
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{\displaystyle f(x)=g(x)h(x)\,\!}
两边取自然对数,可得如下形式的和函数
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{\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
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{\displaystyle \frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}}
整理,可得[7]
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{\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
对如下形式的两个函数的商函数
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{\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
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{\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
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{\displaystyle \frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}-\frac{h'(x)}{h(x)}}
整理,可得
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{\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
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{\displaystyle f(x)}
运用除法定则 所得结果相同。
复合指数函数
对于如下形式的函数
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{\displaystyle f(x)=g(x)^{h(x)}\,\!}
两边取自然对数,可得如下形式的积函数
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{\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
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{\displaystyle \frac{f'(x)}{f(x)} = h'(x) \ln(g(x)) + h(x)\frac{g'(x)}{g(x)}}
整理,得
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{\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。
参见
参考文献
↑ Krantz, Steven G. Calculus demystified. McGraw-Hill Professional. 2003: 170. ISBN 0-07-139308-0 .
↑ N.P. Bali. Golden Differential Calculus. Firewall Media. 2005: 282. ISBN 81-7008-152-1 .
↑ 3.0 3.1 Bird, John. Higher Engineering Mathematics. Newnes. 2006: 324. ISBN 0-7506-8152-7 .
↑ Dowling, Edward T. Schaum's Outline of Theory and Problems of Calculus for Business, Economics, and the Social Sciences. McGraw-Hill Professional. 1990: 160. ISBN 0-07-017673-6 .
↑ Hirst, Keith. Calculus of One Variable. Birkhäuser. 2006: 97. ISBN 1-85233-940-3 .
↑ Blank, Brian E. Calculus, single variable. Springer. 2006: 457. ISBN 1-931914-59-1 .
↑ Williamson, Benjamin. An Elementary Treatise on the Differential Calculus. BiblioBazaar, LLC. 2008: 25–26. ISBN 0-559-47577-2 .
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