導數列表

以下的列表列出了許多函數的導數。f 和g是可微函數，而別的皆為常數。用這些公式，可以求出任何初等函數的導數。

## 一般求導法則

線性法則：$${\mbox{d}(Mf)\over\mbox{d}x}=M{\mbox{d}f\over\mbox{d}x};\qquad [Mf(x)]'=Mf'(x)$$

$${{\mbox{d}(f\pm g)}\over{\mbox{d}x}}={\mbox{d}f\over\mbox{d}x}\pm{\mbox{d}g\over\mbox{d}x}\ $$

乘法定則：$${\mbox{d}fg\over\mbox{d}x}={\mbox{d}f\over\mbox{d}x}g+f\frac{\mbox{d}g}{\mbox{d}x}$$

除法定則：$$\frac{\mbox{d}\dfrac{f}{g}}{\mbox{d}x}= \frac{\dfrac{\mbox{d}f}{\mbox{d}x}g-f\dfrac{\mbox{d}g}{\mbox{d}x}}{g^2}\qquad(g\ne0)$$

倒數定則：$$\frac{\mbox{d}\dfrac{1}{g}}{\mbox{d}x}= \frac{-\dfrac{\mbox{d}g}{\mbox{d}x}}{g^2} \qquad(g\ne0)$$

複合函數求導法則（連鎖定則）：$$(f \circ g)'(x) = f'(g(x)) g'(x).$$

$$\frac{\mbox{d}f[g(x)]}{\mbox{d}x}=\frac{\mbox{d}f(g)}{\mbox{d}g}\frac{\mbox{d}g}{\mbox{d}x}= f'[g(x)]g'(x)$$

反函數的導數：由於 $g(f(x))=x$，故 $g(f(x))'=1$，根據複合函數求導法則，則 $g(f(x))'= \frac{\mbox{d}g[f(x)]}{\mbox{d}x}= \frac{\mbox{d}g(f)}{\mbox{d}f} \frac{\mbox{d}f}{\mbox{d}x}=1$

所以 $\frac{\mbox{d}f}{\mbox{d}x}=\frac{1}{\dfrac{\mbox{d}g(f)}{\mbox{d}f}}=[{\frac{\mbox{d}g(f)}{\mbox{d}f}}]^{-1}= [g'(f)]^{-1}$

同理 $\frac{\mbox{d}g}{\mbox{d}x}=\frac{1}{\dfrac{\mbox{d}f(g)}{\mbox{d}g}}=[{\frac{\mbox{d}f(g)}{\mbox{d}g}}]^{-1}= [f'(g)]^{-1}$

廣義冪法則：$$(f^g)'= \left(e^{g\ln f}\right)' =f^g \left( g'\ln f + \frac{g}{f} f' \right)$$

## 代數函數的導數

(n為任意實常數)：$${\mbox{d}n\over\mbox{d}x}=0$$

$${\mbox{d}x\over\mbox{d}x}=1$$

${\mbox{d}x^n\over\mbox{d}x}=nx^{n-1}\qquad$ 當$n\le1$，則$x\ne0$

$${\mbox{d}|x|\over\mbox{d}x}={x\over|x|}={|x|\over x}=\sgn x\qquad x\ne0$$

## 指數和對數函數的導數

$$\frac{\mbox{d}\ e^{f(x)}}{\mbox{d}x}= f'(x)e^{f(x)}$$

$$\begin{align} \frac{\mbox{d}\ e^x}{\mbox{d}x}&=\lim_{\Delta x\to0}\frac{e^x-e^{x-\Delta x}}{\Delta x}\\ &=e^x\lim_{\Delta x\to0}\frac{1-e^{-\Delta x}}{\Delta x}\\ &=e^x \end{align}$$

$\begin{align} \frac{\mbox{d}\ \alpha^x}{\mbox{d}x}&=\frac{\mbox{d}\ e^{x\!\ln\!\alpha}}{\mbox{d}x}\\ &=\frac{\mbox{d}e^{x\!\ln\!\alpha}}{\mbox{d}\ x\!\ln\!\alpha}\cdot\frac{\mbox{d}\ x\!\ln\!\alpha}{\mbox{d}x}\\ &=e^{x\!\ln\!\alpha}\!\ln\!\alpha\\ &=\alpha^x\!\ln\!\alpha \end{align}$這是前述廣義冪法則在「$f(x)=\alpha$且$g(x)=x$」時的特例。

$$\begin{align} \frac{\mbox{d}\ln x}{\mbox{d}x} &= \lim_{h \to 0} \frac{\ln (x + h) - \ln x}{h} \\ &= \lim_{h \to 0} (\frac{1}{h} \ln (\frac{x + h}{x})) \\ &= \lim_{h \to 0} (\frac{x}{xh} \ln (1 + \frac{h}{x})) \\ &= \frac{1}{x} \ln ( \lim_{h \to 0} (1 + \frac{h}{x})^{\frac{x}{h}}) \\ &= \frac{1}{x} \ln e \\ &= \frac{1}{x} \end{align}$$

$$\frac{\mbox{d}\log_\alpha|x|}{\mbox{d}x}={1\over\ln\alpha}\frac{\mbox{d}\ln|x|}{\mbox{d}x}={1\over x\ln\alpha}$$

$\frac{\mbox{d}\ x^x}{\mbox{d}x}=x^x(1+\ln x)$這是前述廣義冪法則在「$f(x)=x$且$g(x)=x$」時的特例。

## 三角函數的導數

$$\begin{align} (\sin x)' &= \lim_{h \to 0} \frac{\sin(x+h)-\sin x}{h}\\ &= \lim_{h \to 0} \frac{\sin x \cos h + \cos x \sin h - \sin x}{h}\\ &= \lim_{h \to 0} ( \sin x \frac{\cos h - 1}{h} + \cos x \frac{\sin h}{h} )\\ &= \cos x \end{align}$$

$$\begin{align} (\cos x)' &= \lim_{h \to 0} \frac{\cos (x+h)-\cos x}{h}\\ &= \lim_{h \to 0} \frac{\cos x \cos h - \sin x \sin h - \cos x}{h}\\ &= \lim_{h \to 0} ( \cos x \frac{\cos h - 1}{h} - \sin x \frac{\sin h}{h} )\\ &= - \sin x \end{align}$$

$$\begin{align} (\tan x)' &= (\frac{\sin x}{\cos x})' \\ &= \frac{(\sin x)' \cos x - \sin x (\cos x)'}{\cos^2x} \\ &= \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \\ &= \frac{1}{\cos^2 x} = \sec^2 x \end{align}$$

$$\begin{align} (\cot x)' &= (\frac{\cos x}{\sin x})' \\ &= \frac{(\cos x)' \sin x - \cos x (\sin x)'}{\sin^2x} \\ &= \frac{-\sin^2 x - \cos^2 x}{\sin^2 x} \\ &= -\frac{1}{\sin^2 x} = -\csc^2 x \end{align}$$

$$\begin{align} (\sec x)' &= (\frac{1}{\cos x})' \\ &= \frac{\sin x}{\cos^2 x} \\ &= \sec x \tan x \end{align}$$

$$\begin{align} (\csc x)' &= (\frac{1}{\sin x})' \\ &= \frac{-\cos x}{\sin^2 x} \\ &= -\csc x \cot x \end{align}$$

## 反三角函數的導數

$$\begin{align} (\arcsin x)' &= \frac{1}{\cos(\arcsin x)} \Leftrightarrow \sin(\arcsin x) = x \Leftrightarrow \cos(\arcsin x) (\arcsin x)' = 1 \\ &= \frac{1}{\sqrt{1 - \sin^2(\arcsin x)}} \\ &= \frac{1}{\sqrt{1 - x^2}} \ \ (\left| x \right| < 1) \end{align}$$

$$\begin{align} (\arccos x)' &= \frac{1}{-\sin(\arccos x)} \Leftrightarrow \cos(\arccos x) = x \Leftrightarrow -\sin(\arccos x) (\arccos x)' = 1 \\ &= -\frac{1}{\sqrt{1 - \cos^2(\arccos x)}} \\ &= -\frac{1}{\sqrt{1 - x^2}} \ \ (\left| x \right| < 1) \end{align}$$

$$\begin{align} (\arctan x)' &= \frac{1}{\sec^2(\arctan x)} \Leftrightarrow \tan(\arctan x) = x \Leftrightarrow \sec^2(\arctan x) (\arctan x)' = 1 \\ &= \frac{1}{1 + \tan^2(\arctan x)} \\ &= \frac{1}{1 + x^2} \end{align}$$

$$\begin{align} (\arccot x)' &= \frac{1}{-\csc^2(\arccot x)} \Leftrightarrow \cot(\arccot x) = x \Leftrightarrow -\csc^2(\arccot x) (\arccot x)' = 1 \\ &= -\frac{1}{1 + \cot^2(\arccot x)} \\ &= -\frac{1}{1 + x^2} \end{align}$$

$$\begin{align} (\arcsec x)' &= \frac{1}{\sec(\arcsec x)\tan(\arcsec x)} \Leftrightarrow \sec(\arcsec x) = x \Leftrightarrow \sec(\arcsec x)\tan(\arcsec x) (\arcsec x)' = 1 \\ &= \frac{1}{|x| \sqrt{\sec^2(\arcsec x) - 1}} \\ &= \frac{1}{|x| \sqrt{x^2 - 1}} \ \ (\left| x \right| > 1) \end{align}$$

$$\begin{align} (\arccsc x)' &= \frac{1}{-\csc(\arccsc x)\cot(\arccsc x)} \Leftrightarrow \csc(\arccsc x) = x \Leftrightarrow -\csc(\arccsc x)\cot(\arccsc x) (\arccsc x)' = 1 \\ &= -\frac{1}{|x| \sqrt{\csc^2(\arcsec x) - 1}} \\ &= -\frac{1}{|x| \sqrt{x^2 - 1}} \ \ (\left| x \right| > 1) \end{align}$$

## 雙曲函數的導數

$( \sinh x )'= \cosh x = \frac{e^x + e^{-x}}{2}$｜$(\operatorname{arsinh}\,x)' = { 1 \over \sqrt{x^2 + 1}}$

$(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}$｜$(\operatorname{arcosh}\,x)' = { 1 \over \sqrt{x^2 - 1}} (x > 1)$

$(\tanh x )'= \operatorname{sech}^2\,x$｜$(\operatorname{artanh}\,x)' = { 1 \over 1 - x^2} (|x| < 1)$

$(\operatorname{sech}\,x)' = - \tanh x\,\operatorname{sech}\,x$｜$(\operatorname{arsech}\,x)' = -{1 \over x\sqrt{1 - x^2}}(0 < x < 1)$

$(\operatorname{csch}\,x)' = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x(x \neq 0)$｜$(\operatorname{arcsch}\,x)' = -{1 \over |x|\sqrt{1 + x^2}}(x \neq 0)$

$(\operatorname{coth}\,x )' = -\,\operatorname{csch}^2\,x(x \neq 0)$｜$(\operatorname{arcoth}\,x)' = { 1 \over 1 - x^2} (|x| > 1)$

## 特殊函數的導數

伽瑪函數 $\frac{\mbox{d}\Gamma(x)}{\mbox{d}x}=\int^\infty_0e^{-t}t^{x-1}\ln\!t\mbox{d}t$