导数列表

Template:微积分学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

線性法则

$\frac{\text{d}(Mf)}{\text{d}x}=M\frac{\text{d}f}{\text{d}x};[Mf(x){]}^{\prime }=M{f}^{\prime }(x)$

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

乘法定则

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

除法定则

$\frac{\text{d}\frac{f}{g}}{\text{d}x}=\frac{\frac{\text{d}f}{\text{d}x}g-f\frac{\text{d}g}{\text{d}x}}{g^2}(g\ne 0)$

倒数定则

$\frac{\text{d}\frac{1}{g}}{\text{d}x}=\frac{-\frac{\text{d}g}{\text{d}x}}{g^2}(g\ne 0)$

复合函数求导法则（連鎖定則）

$(f\circ g{)}^{\prime }(x)=f^{\prime }(g(x))g^{\prime }(x).$

$\frac{\text{d}f[g(x)]}{\text{d}x}=\frac{\text{d}f(g)}{\text{d}g}\frac{\text{d}g}{\text{d}x}=f^{\prime }[g(x)]g^{\prime }(x)$

反函数的导数

由于 $g(f(x))=x$，故 $g(f(x){)}^{\prime }=1$，根據复合函数求导法则，則 $g(f(x){)}^{\prime }=\frac{\text{d}g[f(x)]}{\text{d}x}=\frac{\text{d}g(f)}{\text{d}f}\frac{\text{d}f}{\text{d}x}=1$

所以 $\frac{\text{d}f}{\text{d}x}=\frac{1}{\frac{\text{d}g(f)}{\text{d}f}}=[\frac{\text{d}g(f)}{\text{d}f}{]}^{-1}=[g^{\prime }(f){]}^{-1}$

同理 $\frac{\text{d}g}{\text{d}x}=\frac{1}{\frac{\text{d}f(g)}{\text{d}g}}=[\frac{\text{d}f(g)}{\text{d}g}{]}^{-1}=[f^{\prime }(g){]}^{-1}$

广义幂法则

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

代数函数的导数

(n为任意实常数)

$\frac{\text{d}n}{\text{d}x}=0$

$\frac{\text{d}x}{\text{d}x}=1$

$\frac{\text{d}{x}^n}{\text{d}x}=n{x}^{n-1}$ 當$n\le 1$，則$x\ne 0$

$\frac{\text{d}|x|}{\text{d}x}=\frac{x}{|x|}=\frac{|x|}{x}=\mathrm{sgn}xx\ne 0$

指数和对数函数的导数

$\frac{\text{d}{e}^{f(x)}}{\text{d}x}=f^{\prime }(x)e^{f(x)}$

$\begin{array}{cc}\frac{\text{d}{e}^x}{\text{d}x}& =\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{e^x-e^{x-\mathrm{\Delta }x}}{\mathrm{\Delta }x}\\ & =e^x\underset{\mathrm{\Delta }x\to 0}{\lim }\frac{1-e^{-\mathrm{\Delta }x}}{\mathrm{\Delta }x}\\ & =e^x\end{array}$

$\begin{array}{cc}\frac{\text{d}{\alpha }^x}{\text{d}x}& =\frac{\text{d}{e}^{x\ln \alpha }}{\text{d}x}\\ & =\frac{\text{d}{e}^{x\ln \alpha }}{\text{d}x\ln \alpha }\cdot \frac{\text{d}x\ln \alpha }{\text{d}x}\\ & =e^{x\ln \alpha }\ln \alpha \\ & ={\alpha }^x\ln \alpha \end{array}$Template:Notetag

$\begin{array}{cc}\frac{\text{d}\ln x}{\text{d}x}& =\underset{h\to 0}{\lim }\frac{\ln (x+h)-\ln x}{h}\\ & =\underset{h\to 0}{\lim }(\frac{1}{h}\ln (\frac{x+h}{x}))\\ & =\underset{h\to 0}{\lim }(\frac{x}{xh}\ln (1+\frac{h}{x}))\\ & =\frac{1}{x}\ln (\underset{h\to 0}{\lim }(1+\frac{h}{x}{)}^{\frac{x}{h}})\\ & =\frac{1}{x}\ln e\\ & =\frac{1}{x}\end{array}$

$\frac{\text{d}{\log }_{\alpha }|x|}{\text{d}x}=\frac{1}{\ln \alpha }\frac{\text{d}\ln |x|}{\text{d}x}=\frac{1}{x\ln \alpha }$

$\frac{\text{d}{x}^x}{\text{d}x}=x^x(1+\ln x)$Template:Notetag

三角函数的导数

$\begin{array}{cc}(\sin x{)}^{\prime }& =\underset{h\to 0}{\lim }\frac{\sin (x+h)-\sin x}{h}\\ & =\underset{h\to 0}{\lim }\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}\\ & =\underset{h\to 0}{\lim }(\sin x\frac{\cos h-1}{h}+\cos x\frac{\sin h}{h})\\ & =\cos x\end{array}$

$\begin{array}{cc}(\cos x{)}^{\prime }& =\underset{h\to 0}{\lim }\frac{\cos (x+h)-\cos x}{h}\\ & =\underset{h\to 0}{\lim }\frac{\cos x\cos h-\sin x\sin h-\cos x}{h}\\ & =\underset{h\to 0}{\lim }(\cos x\frac{\cos h-1}{h}-\sin x\frac{\sin h}{h})\\ & =-\sin x\end{array}$

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

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

$\begin{array}{cc}(\sec x{)}^{\prime }& =(\frac{1}{\cos x}{)}^{\prime }\\ & =\frac{\sin x}{{\cos }^{2}x}\\ & =\sec x\tan x\end{array}$

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

反三角函數的導數

$\begin{array}{cc}(\arcsin x{)}^{\prime }& =\frac{1}{\cos (\arcsin x)}\iff \sin (\arcsin x)=x\iff \cos (\arcsin x)(\arcsin x{)}^{\prime }=1\\ & =\frac{1}{\sqrt{1-{\sin }^2(\arcsin x)}}\\ & =\frac{1}{\sqrt{1-x^2}}(\left|x\right|<1)\end{array}$

$\begin{array}{cc}(\arccos x{)}^{\prime }& =\frac{1}{-\sin (\arccos x)}\iff \cos (\arccos x)=x\iff -\sin (\arccos x)(\arccos x{)}^{\prime }=1\\ & =-\frac{1}{\sqrt{1-{\cos }^2(\arccos x)}}\\ & =-\frac{1}{\sqrt{1-x^2}}(\left|x\right|<1)\end{array}$

$\begin{array}{cc}(\arctan x{)}^{\prime }& =\frac{1}{{\sec }^2(\arctan x)}\iff \tan (\arctan x)=x\iff {\sec }^2(\arctan x)(\arctan x{)}^{\prime }=1\\ & =\frac{1}{1+{\tan }^2(\arctan x)}\\ & =\frac{1}{1+x^2}\end{array}$

$\begin{array}{cc}(\mathrm{arccot}x{)}^{\prime }& =\frac{1}{-{\csc }^2(\mathrm{arccot}x)}\iff \cot (\mathrm{arccot}x)=x\iff -{\csc }^2(\mathrm{arccot}x)(\mathrm{arccot}x{)}^{\prime }=1\\ & =-\frac{1}{1+{\cot }^2(\mathrm{arccot}x)}\\ & =-\frac{1}{1+x^2}\end{array}$

$\begin{array}{cc}(\mathrm{arcsec}x{)}^{\prime }& =\frac{1}{\sec (\mathrm{arcsec}x)\tan (\mathrm{arcsec}x)}\iff \sec (\mathrm{arcsec}x)=x\iff \sec (\mathrm{arcsec}x)\tan (\mathrm{arcsec}x)(\mathrm{arcsec}x{)}^{\prime }=1\\ & =\frac{1}{|x|\sqrt{{\sec }^2(\mathrm{arcsec}x)-1}}\\ & =\frac{1}{|x|\sqrt{x^2-1}}(\left|x\right|>1)\end{array}$

$\begin{array}{cc}(\mathrm{arccsc}x{)}^{\prime }& =\frac{1}{-\csc (\mathrm{arccsc}x)\cot (\mathrm{arccsc}x)}\iff \csc (\mathrm{arccsc}x)=x\iff -\csc (\mathrm{arccsc}x)\cot (\mathrm{arccsc}x)(\mathrm{arccsc}x{)}^{\prime }=1\\ & =-\frac{1}{|x|\sqrt{{\csc }^2(\mathrm{arcsec}x)-1}}\\ & =-\frac{1}{|x|\sqrt{x^2-1}}(\left|x\right|>1)\end{array}$

双曲函数的导数

$(\sinh x{)}^{\prime }=\cosh x=\frac{e^x+e^{-x}}{2}$	$(\mathrm{arsinh}x{)}^{\prime }=\frac{1}{\sqrt{x^2+1}}$
$(\cosh x{)}^{\prime }=\sinh x=\frac{e^x-e^{-x}}{2}$	$(\mathrm{arcosh}x{)}^{\prime }=\frac{1}{\sqrt{x^2-1}}(x>1)$
$(\tanh x{)}^{\prime }={\mathrm{sech}}^{2}x$	$(\mathrm{artanh}x{)}^{\prime }=\frac{1}{1-x^2}(|x|<1)$
$(\mathrm{sech}x{)}^{\prime }=-\tanh x\mathrm{sech}x$	$(\mathrm{arsech}x{)}^{\prime }=-\frac{1}{x\sqrt{1-x^2}}(0<x<1)$
$(\mathrm{csch}x{)}^{\prime }=-\coth x\mathrm{csch}x(x\ne 0)$	$(\mathrm{arcsch}x{)}^{\prime }=-\frac{1}{|x|\sqrt{1+x^2}}(x\ne 0)$
$(\coth x{)}^{\prime }=-{\mathrm{csch}}^{2}x(x\ne 0)$	$(\mathrm{arcoth}x{)}^{\prime }=\frac{1}{1-x^2}(|x|>1)$

特殊函数的导数

伽玛函数 $\frac{\text{d}\mathrm{\Gamma }(x)}{\text{d}x}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}{t}^{x-1}\ln t\text{d}t$