查看“︁导数列表”︁的源代码

{{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
:<math>{\mbox{d}(Mf)\over\mbox{d}x}=M{\mbox{d}f\over\mbox{d}x};\qquad [Mf(x)]'=Mf'(x)</math>
:<math>{{\mbox{d}(f\pm g)}\over{\mbox{d}x}}={\mbox{d}f\over\mbox{d}x}\pm{\mbox{d}g\over\mbox{d}x}\ </math>
;[[乘法定则]]
:<math>{\mbox{d}fg\over\mbox{d}x}={\mbox{d}f\over\mbox{d}x}g+f\frac{\mbox{d}g}{\mbox{d}x}</math>
;[[除法定则]]
:<math>\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)</math>
;[[倒数定则]]
:<math>\frac{\mbox{d}\dfrac{1}{g}}{\mbox{d}x}= \frac{-\dfrac{\mbox{d}g}{\mbox{d}x}}{g^2} \qquad(g\ne0) </math>
;[[复合函数求导法则]]（連鎖定則）
:<math>(f \circ g)'(x) = f'(g(x)) g'(x).</math>
:<math>\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)</math>
;[[反函数]]的导数
:由于 <math>g(f(x))=x</math>，故 <math>g(f(x))'=1</math>，根據复合函数求导法则，則 <math> 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</math>
:所以 <math> \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}</math>
:同理 <math> \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}</math>
;广义幂法则
:<math>(f^g)'= \left(e^{g\ln f}\right)' =f^g \left( g'\ln f + \frac{g}{f} f' \right)</math>

==[[代数]]函数的导数==
;(n为任意实常数)
:<math>{\mbox{d}n\over\mbox{d}x}=0</math>
:<math>
{\mbox{d}x\over\mbox{d}x}=1</math>
:<math>{\mbox{d}x^n\over\mbox{d}x}=nx^{n-1}\qquad</math> 當<math>n\le1</math>，則<math>x\ne0</math>
:<math>{\mbox{d}|x|\over\mbox{d}x}={x\over|x|}={|x|\over x}=\sgn x\qquad x\ne0</math>

==[[指数函数|指数]]和[[对数]]函数的导数==
:<math>
\frac{\mbox{d}\ e^{f(x)}}{\mbox{d}x}= f'(x)e^{f(x)}
</math>

:<math>
\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}
</math>

:<math>
\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}
</math>{{notetag|這是前述廣義冪法則在「<math>f(x)=\alpha</math>且<math>g(x)=x</math>」時的特例。}}

:<math>
\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}
</math>

:<math>\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}</math>

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


==[[三角函数]]的导数==
<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}

</math>

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

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

==[[反三角函數]]的導數==
<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

<math>
\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}
</math>

==[[双曲函数]]的导数==
{| style="width:100%; background:transparent; margin-left:2em;"
|width=50%|<math>( \sinh x )'= \cosh x = \frac{e^x + e^{-x}}{2}</math>
|width=50%|<math>(\operatorname{arsinh}\,x)' = { 1 \over \sqrt{x^2 + 1}}</math>
|-
|<math>(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}</math>
|<math>(\operatorname{arcosh}\,x)' = { 1 \over \sqrt{x^2 - 1}} (x > 1)</math>
|-
|<math>(\tanh x )'= \operatorname{sech}^2\,x</math>
|<math>(\operatorname{artanh}\,x)' = { 1 \over 1 - x^2} (|x| < 1)</math>
|-
|<math>(\operatorname{sech}\,x)' = - \tanh x\,\operatorname{sech}\,x</math>
|<math>(\operatorname{arsech}\,x)' = -{1 \over x\sqrt{1 - x^2}}(0 < x < 1)</math>
|-
|<math>(\operatorname{csch}\,x)' = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x(x \neq 0)</math>
|<math>(\operatorname{arcsch}\,x)' = -{1 \over |x|\sqrt{1 + x^2}}(x \neq 0)</math>
|-
|<math>(\operatorname{coth}\,x )' = -\,\operatorname{csch}^2\,x(x \neq 0)</math>
|<math>(\operatorname{arcoth}\,x)' = { 1 \over 1 - x^2} (|x| > 1)</math>
|}

==[[特殊函数]]的导数==
{| style="width:100%; background:transparent; margin-left:2em;"
|width=50%|
;[[伽玛函数]]
<math>\frac{\mbox{d}\Gamma(x)}{\mbox{d}x}=\int^\infty_0e^{-t}t^{x-1}\ln\!t\mbox{d}t</math>
|width=50%|
|}

==註釋==
{{notefoot}}

[[Category:包含证明的条目]]
[[Category:求导法则| ]]
[[Category:微分学|*]]
[[Category:数学列表|Derivatives]]
[[Category:数学用表|Derivatives]]
[[Category:数学恒等式]]
[[Category:分析定理]]
[[Category:微積分定理]]