倒数定则此条目没有列出任何参考或来源。 (2009年2月6日)维基百科所有的内容都应该可供查证。请协助补充可靠来源以改善这篇条目。无法查证的内容可能会因为异议提出而移除。倒数定则(英语: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 ) {\displaystyle g(x)} ,则其倒数 1 g ( x ) {\displaystyle {\frac {1}{g(x)}}} 的导数为 d d x ( 1 g ( x ) ) = − g ′ ( x ) ( g ( x ) ) 2 {\displaystyle {\frac {d}{dx}}\left({\frac {1}{g(x)}}\right)={\frac {-g'(x)}{(g(x))^{2}}}} 例子 1 x 2 + 2 x {\displaystyle {\frac {1}{x^{2}+2x}}} 的导数为: d d x ( 1 x 2 + 2 x ) = − 2 x − 2 ( x 2 + 2 x ) 2 . {\displaystyle {\frac {d}{dx}}\left({\frac {1}{x^{2}+2x}}\right)={\frac {-2x-2}{(x^{2}+2x)^{2}}}.} 1 cos ( x ) {\displaystyle {\frac {1}{\cos(x)}}} 的导数为: d d x ( 1 cos ( x ) ) = sin ( x ) cos 2 ( x ) = 1 cos ( x ) sin ( x ) cos ( x ) = sec ( x ) tan ( x ) . {\displaystyle {\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 {\displaystyle f(x)=1} ,则根据除法定则可得 d d x ( 1 g ( x ) ) = d d x ( f ( x ) g ( x ) ) {\displaystyle {\frac {d}{dx}}\left({\frac {1}{g(x)}}\right)={\frac {d}{dx}}\left({\frac {f(x)}{g(x)}}\right)} = f ′ ( x ) g ( x ) − f ( x ) g ′ ( x ) ( g ( x ) ) 2 {\displaystyle ={\frac {f'(x)g(x)-f(x)g'(x)}{(g(x))^{2}}}} = 0 ⋅ g ( x ) − 1 ⋅ g ′ ( x ) ( g ( x ) ) 2 {\displaystyle ={\frac {0\cdot g(x)-1\cdot g'(x)}{(g(x))^{2}}}} = − g ′ ( x ) ( g ( x ) ) 2 . {\displaystyle ={\frac {-g'(x)}{(g(x))^{2}}}.}