Question

设函数 y=f(x)的导数 f'(x)与二阶导数f''(x) 存在且均不为零,其反函数为x=φ(y) ,则φ''(y)

Answer 1

u'(y)=1/f'(x)=1/f'(u(y))

u''(y)=(1/f'(u(y)))'=-1/(f'(x))^2 * f''(x) * u‘(y) （复合函数求导）

=-f''(x)/(f'(x))^2 * 1/f'(x)

=-f''(x)/(f'(x))^3