设y=f(x)的一价,二价导数存在且为非零,其反函数为x=g(y),证明:g''(y)=-f''(x)/[f'(x)]^3

问题描述:

设y=f(x)的一价,二价导数存在且为非零,其反函数为x=g(y),证明:g''(y)=-f''(x)/[f'(x)]^3

因为g'(y)=1/f'(x)=1/f'(g(y))
故根据复合函数求导得(注意y是自变量)
g''(y)=-f''(g(y))/f'²(g(y))*g'(y)=-f''(g(y))/[f'(g(y))]^3=-f''(x)/[f'(x)]^3