查看“︁对数微分法”︁的源代码

{{微积分学}}
'''对数微分法'''（{{lang-en|Logarithmic differentiation}}）是在[[微积分学]]中，通过求某[[函数]]''f''的{{link-en|对数导数|Logarithmic derivative}}来求得函数[[导数]]的一种方法， <ref>{{cite book|title=Calculus demystified|url=https://archive.org/details/calculusdemystif0000kran|pages=[https://archive.org/details/calculusdemystif0000kran/page/170 170]|first=Steven G.|last=Krantz|publisher=McGraw-Hill Professional|year=2003|isbn=0-07-139308-0}}</ref>

:<math>[\ln(f)]' = \frac{f'}{f} \quad \rightarrow \quad  f' = f \cdot [\ln(f)]'.</math>

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<ref>{{cite book|title=Golden Differential Calculus|pages=282|author=N.P. Bali|publisher=Firewall Media|year=2005|isbn=81-7008-152-1}}</ref><ref name="Bird">{{cite book|title=Higher Engineering Mathematics|url=https://archive.org/details/higherengineerin00bscj_519|first=John|last=Bird|pages=[https://archive.org/details/higherengineerin00bscj_519/page/n342 324]|publisher=Newnes|year=2006|isbn=0-7506-8152-7}}</ref>。这一方法可以应用于所有恆不为0的[[可微函数]]。

==概述 ==
对于某函数

:<math>y=f(x)\,\!</math>

运用对数微分法，通常对函数两边取绝对值后取自然对数<ref>{{cite book|title=Schaum's Outline of Theory and Problems of Calculus for Business, Economics, and the Social Sciences|url=https://archive.org/details/schaumsoutlineof0000dowl|first=Edward T.|last=Dowling|publisher=McGraw-Hill Professional|year=1990|isbn=0-07-017673-6|pages=[https://archive.org/details/schaumsoutlineof0000dowl/page/160 160]}}</ref>。

:<math>\ln|y| = \ln|f(x)|\,\!</math>

运用[[隐函数#隐函数的导数|隐式微分法]]<ref name="One">{{cite book|title=Calculus of One Variable|first=Keith|last=Hirst|pages=97|publisher=Birkhäuser|year=2006|isbn=1-85233-940-3}}</ref>，可得

:<math>\frac{1}{y} \frac{dy}{dx} = \frac{f'(x)}{f(x)}</math>

两边同乘以''y''，则方程左边只剩下''dy''/''dx''：

:<math>\frac{dy}{dx} = y \times \frac{f'(x)}{f(x)} = f'(x).</math>

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<ref>{{cite book|title=Calculus, single variable|url=https://archive.org/details/calculussingleva0000blan|first=Brian E.|last=Blank|pages=[https://archive.org/details/calculussingleva0000blan/page/457 457]|publisher=Springer|year=2006|isbn=1-931914-59-1}}</ref>，常用的对数性质有：<ref name="Bird" />

: <math>\ln(ab) = \ln(a) + \ln(b), \qquad
\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b), \qquad 
\ln(a^n) = n\ln(a)</math>

===通用公式===
有一如下形式的函数，
:<math>f(x)=\prod_i(f_i(x))^{\alpha_i(x)}.</math>
两边取自然对数，得
:<math>\ln (f(x))=\sum_i\alpha_i(x)\cdot \ln(f_i(x)),</math>
两边对''x''求导，得
:<math>\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].</math>
两边同乘以<math>f(x)</math>，可得原函数的导数为
:<math>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))]'}</math>

==应用==

===积函数===
对如下形式的两个函数的积函数
:<math>f(x)=g(x)h(x)\,\!</math>
两边取自然对数，可得如下形式的和函数
:<math>\ln(f(x))=\ln(g(x)h(x))=\ln(g(x))+\ln(h(x))\,\!</math>
应用链式法则，两边微分，得
:<math>\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}+\frac{h'(x)}{h(x)}</math>
整理，可得<ref>{{cite book|title=An Elementary Treatise on the Differential Calculus|first=Benjamin|last=Williamson|publisher=BiblioBazaar, LLC|year=2008|pages=25–26|isbn=0-559-47577-2}}</ref>
:<math>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\}</math>

===商函数===
对如下形式的两个函数的商函数
:<math>f(x)=\frac{g(x)}{h(x)}\,\!</math>
两边取自然对数，可得如下形式的差函数
:<math>\ln(f(x))=\ln\Bigg(\frac{g(x)}{h(x)}\Bigg)=\ln(g(x))-\ln(h(x))\,\!</math>
应用链式法则，两边求导，得
:<math>\frac{f'(x)}{f(x)} = \frac{g'(x)}{g(x)}-\frac{h'(x)}{h(x)}</math>
整理，可得
:<math>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\}</math>

右边通分之后，结果和对<math>f(x)</math>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
:<math>f(x)=g(x)^{h(x)}\,\!</math>
两边取自然对数，可得如下形式的积函数
:<math>\ln(f(x))=\ln\left(g(x)^{h(x)}\right)=h(x) \ln(g(x))\,\!</math>
应用链式法则，两边求导，得
:<math>\frac{f'(x)}{f(x)} = h'(x) \ln(g(x)) + h(x)\frac{g'(x)}{g(x)}</math>
整理，得
:<math>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\}.</math>
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==
{{Portal|数学}}
* [[对数恒等式]]

==参考文献==
{{reflist|2}}

==外部链接==
*{{cite web|url=http://v.163.com/movie/2006/8/S/A/M6GLI5A07_M6GLO3OSA.html|title=网易公开课：对数微分法|publisher=网易|accessdate=2014-11-26|archive-date=2020-01-07|archive-url=https://web.archive.org/web/20200107215232/http://v.163.com/movie/2006/8/S/A/M6GLI5A07_M6GLO3OSA.html|dead-url=no}}
*{{cite web|url=https://www.youtube.com/playlist?list=PL89B4CC8694E23F00|title=对数之微分法（高中文理科）|publisher=Youtube|accessdate=2014-11-26|archive-date=2016-03-14|archive-url=https://web.archive.org/web/20160314173610/https://www.youtube.com/playlist?list=PL89B4CC8694E23F00|dead-url=no}}
*{{cite web|url=http://www.mathcentre.ac.uk/students/topics/differentiation/by-logs/|title=Differentiation by taking logarithms – Teach yourself|publisher=mathcentre.ac.uk|accessdate=2012-01-03|archive-date=2020-10-26|archive-url=https://web.archive.org/web/20201026002148/https://mathcentre.ac.uk/students/topics/differentiation/by-logs/|dead-url=no}}
*{{cite web|url=http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/logdiffdirectory/LogDiff.html|title=Logarithmic differentiation|accessdate=2009-03-10|archive-date=2020-11-27|archive-url=https://web.archive.org/web/20201127194305/https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/logdiffdirectory/LogDiff.html|dead-url=no}}
*{{cite web|url=http://tutorial.math.lamar.edu/Classes/CalcI/LogDiff.aspx|title=Calculus I – Logarithmic differentiation|accessdate=2009-03-10|archive-date=2021-01-03|archive-url=https://web.archive.org/web/20210103102757/https://tutorial.math.lamar.edu/Classes/CalcI/LogDiff.aspx|dead-url=no}}

[[Category:微分学]]