函数求导简介

求导公式

$(x^n{)}^{\prime }=n{x}^{n-1}$
$(e^x{)}^{\prime }=e^x$
$(\ln x{)}^{\prime }=\frac{1}{x}$
$(a^x)=a^x\ln a$
$(lo{g}_{a}x{)}^{\prime }=\frac{1}{x\ln a}$


$(\sin x{)}^{\prime }=\cos x$
$(\cos x{)}^{\prime }=\sin x$
$(\tan x{)}^{\prime }={\sec }^{2}x$
$(\cot x{)}^{\prime }=-{\csc }^{2}x$
$(\sec x{)}^{\prime }=\sec x\tan x$
$(\csc x{)}^{\prime }=-\csc x\cot x$


$(\arcsin x{)}^{\prime }=\frac{1}{\sqrt{1-x^2}}$

$(\arccos x{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}$

$(\arctan x{)}^{\prime }=\frac{1}{1+x^2}$

$(\text{arccot}x{)}^{\prime }=-\frac{1}{1+x^2}$

有关三角函数

$\sin x\csc x=1$
$\cos x\sec x=1$
$\tan x\cot x=1$

求导法则

$[f(x)\pm g(x){]}^{\prime }=f^{\prime }(x)\pm g^{\prime }(x)$

$[f(x)\cdot g(x){]}^{\prime }=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

$[\frac{f(x)}{g(x)}{]}^{\prime }=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{g^2(x)}$