對數微分法 (英語:Logarithmic differentiation )是在微積分學 中,通過求某函數 f 的對數導數 來求得函數導數 的一種方法, [1]
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{\displaystyle [\ln(f)]' = \frac{f'}{f} \quad \rightarrow \quad f' = f \cdot [\ln(f)]'.}
這一方法常在函數對數求導比對函數本身求導更容易時使用,這樣的函數通常是幾項的積,取對數之後,可以把函數變成容易求導的幾項的和。這一方法對冪函數形式的函數也很有用。對數微分法依賴於鏈式法則 和對數 的性質(尤其是自然對數 ),把積變為求和,把商變為做差[2] [3] 。這一方法可以應用於所有恆不為0的可微函數 。
概述
對於某函數
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{\displaystyle y=f(x)\,\!}
運用對數微分法,通常對函數兩邊取絕對值後取自然對數[4] 。
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{\displaystyle \ln|y| = \ln|f(x)|\,\!}
運用隱式微分法 [5] ,可得
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{\displaystyle \frac{1}{y} \frac{dy}{dx} = \frac{f'(x)}{f(x)}}
兩邊同乘以y ,則方程左邊只剩下dy /dx :
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{\displaystyle \frac{dy}{dx} = y \times \frac{f'(x)}{f(x)} = f'(x).}
對數微分法有用,是因為對數的性質可以大大簡化複雜函數的微分[6] ,常用的對數性質有:[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
有一如下形式的函數,
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{\displaystyle f(x)=\prod_i(f_i(x))^{\alpha_i(x)}.}
兩邊取自然對數,得
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{\displaystyle \ln (f(x))=\sum_i\alpha_i(x)\cdot \ln(f_i(x)),}
兩邊對x 求導,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].}
兩邊同乘以
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{\displaystyle f(x)}
,可得原函數的導數為
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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))]'}}
應用
積函數
對如下形式的兩個函數的積函數
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{\displaystyle f(x)=g(x)h(x)\,\!}
兩邊取自然對數,可得如下形式的和函數
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{\displaystyle \ln(f(x))=\ln(g(x)h(x))=\ln(g(x))+\ln(h(x))\,\!}
應用鏈式法則,兩邊微分,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\}}
商函數
對如下形式的兩個函數的商函數
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{\displaystyle f(x)=\frac{g(x)}{h(x)}\,\!}
兩邊取自然對數,可得如下形式的差函數
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{\displaystyle \ln(f(x))=\ln\Bigg(\frac{g(x)}{h(x)}\Bigg)=\ln(g(x))-\ln(h(x))\,\!}
應用鏈式法則,兩邊求導,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\}}
右邊通分之後,結果和對
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{\displaystyle f(x)}
運用除法定則 所得結果相同。
複合指數函數
對於如下形式的函數
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{\displaystyle f(x)=g(x)^{h(x)}\,\!}
兩邊取自然對數,可得如下形式的積函數
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{\displaystyle \ln(f(x))=\ln\left(g(x)^{h(x)}\right)=h(x) \ln(g(x))\,\!}
應用鏈式法則,兩邊求導,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\}.}
與將函數f 看做指數函數 ,直接運用鏈式法則所得結果相同。
參見
參考文獻
↑ 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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