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]
![{\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)
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。
對於某函數
![{\displaystyle y=f(x)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f35d62b1a95cbaa7eca6461a3e4944d5fca5d7fe)
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]。
![{\displaystyle \ln |y|=\ln |f(x)|\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd8e53a43d961d81106a98d1c127ac49e0c024f5)
運用隱式微分法[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)
兩邊同乘以y,則方程左邊只剩下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)
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]
![{\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)
有一如下形式的函數,
![{\displaystyle f(x)=\prod _{i}(f_{i}(x))^{\alpha _{i}(x)}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/469708453aa6fc4dd42da542002017a0f3afaad9)
兩邊取自然對數,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)
兩邊對x求導,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)
兩邊同乘以
,可得原函數的導數為
![{\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)
對如下形式的兩個函數的積函數
![{\displaystyle f(x)=g(x)h(x)\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ca7d8f0390f47cdff12e5fd8f7d75ecaa2245f)
兩邊取自然對數,可得如下形式的和函數
![{\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)
應用鏈式法則,兩邊微分,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)
對如下形式的兩個函數的商函數
![{\displaystyle f(x)={\frac {g(x)}{h(x)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5407735d362d45878d666bc1d293a23162e71a0)
兩邊取自然對數,可得如下形式的差函數
![{\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)
應用鏈式法則,兩邊求導,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)
右邊通分之後,結果和對
運用除法定則所得結果相同。
對於如下形式的函數
![{\displaystyle f(x)=g(x)^{h(x)}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e868781839fbea515f80fd8efde1ebefc7c19227)
兩邊取自然對數,可得如下形式的積函數
![{\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)
應用鏈式法則,兩邊求導,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)
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。
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