根据导数定义,f(x)可导即为以下极限存在：
lim(h->0) (f(x+h)-f(x))/h
而：
lim(h->0) (f(x+h)-f(x))/h = lim(h->0) (f(x)g(h)+f(h)g(x)-f(x))/h
= lim(h->0) f(x)(g(h)-1)/h + lim(h->0) f(h)g(x)/h
= f(x)lim(h->0) (g(h)-1)/h + g(x)lim(h->0) f(h)/h
其中：
(1)
lim(h->0) (g(h)-1)/h = lim(h->0) (g(h)-g(0))/h = g'(0) = 0
(2)
lim(h->0) f(h)/h =lim(h->0) (f(h)-0)/h = lim(h->0) (f(h)-f(0))/h = f'(0) = 1
所以
lim(h->0) (f(x+h)-f(x))/h = f(x)*0+g(x)*1 = g(x)
即极限存在且等于g(x),因此f'(x)存在且等于g(x)