导数列表

以下的列表列出了许多函数的导数。f和g是可微函数，而别的皆为常数。用这些公式，可以求出任何初等函数的导数。

一般求导法则

线性：

${\mbox{d}Mf\over\mbox{d}x}=M{\mbox{d}f\over\mbox{d}x}$

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

乘法定则：

${\mbox{d}fg\over\mbox{d}x}={\mbox{d}f\over\mbox{d}x}g+f\frac{\mbox{d}g}{\mbox{d}x}$

倒数定则：

$\frac{\mbox{d}\dfrac{1}{f}}{\mbox{d}x}=\frac{-\dfrac{\mbox{d}f}{\mbox{d}x}}{f^2}$

除法定则：

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

复合函数求导法则：

$(f \circ g)' = (f' \circ g)g'$

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

反函数的导数：

$\begin{align} {\mbox{d}y\over\mbox{d}y}&={\mbox{d}y\over\mbox{d}x}{\mbox{d}x\over\mbox{d}y}&=1\\ {\mbox{d}y\over\mbox{d}x}={1\over{\mbox{d}x\over\mbox{d}y}} \end{align}$

广义幂法则：

$(f^g)'=f^g \left( g'\ln f + \frac{g}{f} f' \right)$

代数函数的导数

(n为任意实常数)

${\mbox{d}M\over\mbox{d}x}=0$

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

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

指数和对数函数的导数

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

$\begin{align} \frac{\mbox{d}\ \alpha^x}{\mbox{d}x}&=\frac{\mbox{d}\ e^{x\!\ln\!\alpha}}{\mbox{d}x}\\ &=\frac{\mbox{d}e^{x\!\ln\!\alpha}}{\mbox{d}\ x\!\ln\!\alpha}\cdot\frac{\mbox{d}\ x\!\ln\!\alpha}{\mbox{d}x}\\ &=e^{x\!\ln\!\alpha}\!\ln\!\alpha\\ &=\alpha^x\!\ln\!\alpha \end{align}$

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

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

$\frac{dx^x}{dx}=x^x(1+\ln x)$

三角函数的导数

$(\sin x)' = \cos x \,$	$(\arcsin x)' = { 1 \over \sqrt{1 - x^2}} \,$
$(\cos x)' = -\sin x \,$	$(\arccos x)' = -{1 \over \sqrt{1 - x^2}} \,$
$(\tan x)' = \sec^2 x = { 1 \over \cos^2 x} \,$	$(\arctan x)' = { 1 \over 1 + x^2} \,$
$(\sec x)' = \sec x \tan x \,$	$(\arcsec x)' = { 1 \over |x|\sqrt{x^2 - 1}} \,$
$(\csc x)' = -\csc x \cot x \,$	$(\arccsc x)' = -{1 \over |x|\sqrt{x^2 - 1}} \,$
$(\cot x)' = -\csc^2 x = -{ 1 \over \sin^2 x} \,$	$(\arccot x)' = -{1 \over 1 + x^2} \,$

双曲函数的导数

$( \sinh x )'= \cosh x = \frac{e^x + e^{-x}}{2}$	$(\operatorname{arsinh}\,x)' = { 1 \over \sqrt{x^2 + 1}}$
$(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}$	$(\operatorname{arcosh}\,x)' = { 1 \over \sqrt{x^2 - 1}} (x > 1)$
$(\tanh x )'= \operatorname{sech}^2\,x$	$(\operatorname{artanh}\,x)' = { 1 \over 1 - x^2} (|x| < 1)$
$(\operatorname{sech}\,x)' = - \tanh x\,\operatorname{sech}\,x$	$(\operatorname{arsech}\,x)' = {-1 \over x\sqrt{1 - x^2}}$
$(\operatorname{csch}\,x)' = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x$	$(\operatorname{arcsch}\,x)' = -{1 \over |x|\sqrt{1 + x^2}}$
$(\operatorname{coth}\,x )' = -\,\operatorname{csch}^2\,x$	$(\operatorname{arcoth}\,x)' = { 1 \over 1 - x^2} (|x| > 1)$

特殊函数的导数

伽玛函数 $\frac{\mbox{d}\Gamma(x)}{\mbox{d}x}=\int^\infty_0e^{-t}t^{x-1}\ln\!t\mbox{d}t$