倒数定则

倒数定则（英語：Reciprocal rule）是数学中关于函数的倒数的导数的一个计算定则。

设有函数$g(x)$，则其倒数${\frac {1}{g(x)}}$的导数为

${\frac {d}{dx}}\left({\frac {1}{g(x)}}\right)={\frac {-g'(x)}{(g(x))^{2}}}$

例子

${\frac {1}{x^{2}+2x}}$的导数为：

${\frac {d}{dx}}\left({\frac {1}{x^{2}+2x}}\right)={\frac {-2x-2}{(x^{2}+2x)^{2}}}.$

${\frac {1}{\cos(x)}}$的导数为：

${\frac {d}{dx}}\left({\frac {1}{\cos(x)}}\right)={\frac {\sin(x)}{\cos ^{2}(x)}}={\frac {1}{\cos(x)}}{\frac {\sin(x)}{\cos(x)}}=\sec(x)\tan(x).$

证明

设 $f(x)=1$，则根据除法定则可得

${\frac {d}{dx}}\left({\frac {1}{g(x)}}\right)={\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)$	$={\frac {f'(x)g(x)-f(x)g'(x)}{(g(x))^{2}}}$
	$={\frac {0\cdot g(x)-1\cdot g'(x)}{(g(x))^{2}}}$
	$={\frac {-g'(x)}{(g(x))^{2}}}.$