对数微分法(英語:Logarithmic differentiation)是在微积分学中,通过求某函数f的对数导数来求得函数导数的一种方法, [1]
![{\displaystyle [\ln(f)]'={\frac {f'}{f}}\quad \rightarrow \quad f'=f\cdot [\ln(f)]'.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b7de91cd8aa179e45a630cef367c78239b62b3)
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恆不为0的可微函数。
对于某函数
![{\displaystyle y=f(x)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f35d62b1a95cbaa7eca6461a3e4944d5fca5d7fe)
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]。
![{\displaystyle \ln |y|=\ln |f(x)|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd8e53a43d961d81106a98d1c127ac49e0c024f5)
运用隐式微分法[5],可得
![{\displaystyle {\frac {1}{y}}{\frac {dy}{dx}}={\frac {f'(x)}{f(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f33b18510ade9875164b040f2d23139e4b4c39)
两边同乘以y,则方程左边只剩下dy/dx:
![{\displaystyle {\frac {dy}{dx}}=y\times {\frac {f'(x)}{f(x)}}=f'(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce94451dc8fd16cbfd47609af2552db89c96f03c)
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]
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/717eaa3e03191166528363499e4a65a69a3a589c)
通用公式[编辑]
有一如下形式的函数,
![{\displaystyle f(x)=\prod _{i}(f_{i}(x))^{\alpha _{i}(x)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/469708453aa6fc4dd42da542002017a0f3afaad9)
两边取自然对数,得
![{\displaystyle \ln(f(x))=\sum _{i}\alpha _{i}(x)\cdot \ln(f_{i}(x)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7e8f47ee57b0584bef2172796be61eab82f65d)
两边对x求导,得
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e36c147dcca7d2378a3e0e964ca18568639d44fa)
两边同乘以
,可得原函数的导数为
![{\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))]'}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72623945515101eff3fc5880c909c33c56e63aba)
积函数[编辑]
对如下形式的两个函数的积函数
![{\displaystyle f(x)=g(x)h(x)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ca7d8f0390f47cdff12e5fd8f7d75ecaa2245f)
两边取自然对数,可得如下形式的和函数
![{\displaystyle \ln(f(x))=\ln(g(x)h(x))=\ln(g(x))+\ln(h(x))\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c7e1089f412ebc709b2ada6eaf5048e8bbd346b)
应用链式法则,两边微分,得
![{\displaystyle {\frac {f'(x)}{f(x)}}={\frac {g'(x)}{g(x)}}+{\frac {h'(x)}{h(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8f960bb80ef1e729bf9631fb227286560dc1fd3)
整理,可得[7]
![{\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 \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/518ddf1df627acec2a83824729ac467adbc36503)
商函数[编辑]
对如下形式的两个函数的商函数
![{\displaystyle f(x)={\frac {g(x)}{h(x)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5407735d362d45878d666bc1d293a23162e71a0)
两边取自然对数,可得如下形式的差函数
![{\displaystyle \ln(f(x))=\ln {\Bigg (}{\frac {g(x)}{h(x)}}{\Bigg )}=\ln(g(x))-\ln(h(x))\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf9a6348fef6eee6587cc2713dd48767cfa025a3)
应用链式法则,两边求导,得
![{\displaystyle {\frac {f'(x)}{f(x)}}={\frac {g'(x)}{g(x)}}-{\frac {h'(x)}{h(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ba926ce56fc0decfd505c6a9bdfb41465d25e7)
整理,可得
![{\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 \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8033618e7b15e159adf1010cc689a9df549f1e8)
右边通分之后,结果和对
运用除法定则所得结果相同。
复合指数函数[编辑]
对于如下形式的函数
![{\displaystyle f(x)=g(x)^{h(x)}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e868781839fbea515f80fd8efde1ebefc7c19227)
两边取自然对数,可得如下形式的积函数
![{\displaystyle \ln(f(x))=\ln \left(g(x)^{h(x)}\right)=h(x)\ln(g(x))\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b077dad0a95ebb7a6655526f2cf9df66f1271ae7)
应用链式法则,两边求导,得
![{\displaystyle {\frac {f'(x)}{f(x)}}=h'(x)\ln(g(x))+h(x){\frac {g'(x)}{g(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/596ffdd575b5313b2662536e1a0cda5cec47ab91)
整理,得
![{\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 \}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb07e0113ac33b414ab15ca1a14bafb743857993)
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。
参考文献[编辑]
- ^ 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.
外部链接[编辑]