设f(x)在点x=0处可导,且f(0)=0,f'(0)不等于0又F(x)在点x=0处亦可导.证明F[f(x)]在点x=0处可导_数学解答

设f(x)在点x=0处可导,且f(0)=0,f'(0)不等于0又F(x)在点x=0处亦可导.证明F[f(x)]在点x=0处可导

人气：361 ℃　时间：2020-06-03 06:10:16

解答

F(x)在点x=0处可导,即当x→0时,lim(F(x)-F(0))/x 存在
由于f(x)在点x=0处可导,必定在x=0处连续,当x→0时,limf(x)=f(0）=0
当x→0时：lim(F[f(x)]-F[f(0)])/x=lim{(F[f(x)]-F[f(0)])/f(x)}{[f(x)-f(0)}/x=F'(0)f'(0)