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
;線性法则
:{\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}\
;乘法定则
:{\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)
;倒数定则
:\frac{\mbox{d}\dfrac{1}{g}}{\mbox{d}x}= \frac{-\dfrac{\mbox{d}g}{\mbox{d}x}}{g^2} \qquad(g\ne0)
;复合函数求导法则（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)
;反函数的导数
:由于 g(f(x))=x，故 g(f(x))'=1，根據复合函数求导法则，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}
;广义幂法则
:(f^g)'= \left(e^{g\ln f}\right)' =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}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|}={\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)

三角函数的导数
\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|

\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|

\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}

特殊函数的导数
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