设y=f(x)在某开区间内二阶可导,导数f'(x)不等0.若y=f(x)的反函数为x=g(y),用f(x)来表示g''(y).

Question

设y=f(x)在某开区间内二阶可导,导数f'(x)不等0.若y=f(x)的反函数为x=g(y),
用f(x)来表示g''(y).

Answer 1

　　已知
　　g'(y) = 1/f'(x),
则
　　g"(y) = (d/dy)g'(y)
　　　 　= (d/dx)[1/f'(x)]*(dx/dy)
　　　 　= {-f'(x)/[f"(x)]^2}*[1/f'(x)]
　　　 　= -f'(x)/[f"(x)]^3.