如果函数x=f(x)x=f(x)x=f(x)在区间III内单调可导, 且f′(y)≠0f'(y)\neq 0f′(y)=0, 则反函数y=f−1(x)y=f^{-1}(x)y=f−1(x)在III也可导, 且
[f−1(x)]′=1f′(y) [f^{-1}(x)]'=\frac{1}{f'(y)} [f−1(x)]′=f′(y)1
根据这个法则，可推导出反三角函数的导数： 设 y=arcsin(x)y=arcsin(x)y=arcsin(x) ,则 x=sin(y)x=sin(y)x=sin(y), 求y′y'y′过程如下
(arcsin(x))′=1sin(y)′=1cos(y)=11−sin2(y)=11−x2 \big(arcsin(x)\big)'=\frac{1}{sin(y)'}=\frac{1}{cos(y)}=\frac{1}{\sqrt{1-sin^2(y)}}=\frac{1}{\sqrt{1-x^2}} (arcsin(x))′=sin(y)′1=cos(y)1=1−sin2(y)1=1−x21
同理, 可推导出arccos(x), arctan(x). 计算时注意下定义域和值域
