题目

f(x)为非0函数高数f(x+y)=f(x)f(y) 当x=0时的导数为1证明f(x)的导数等于f(x）

提问时间：2020-06-13

答案

f(x+y)=f(x)f(y)
put x=y=0
f(0)=f(0)f(0)
f(0) =1
f'(x) = lim(y->0){ [f(x+y)-f(x)]/y}
= lim(y->0) [f(x)f(y)-f(x)]/y
= f(x) lim(y->0)(f(y)-1)/y
= f(x) lim(y->0)( f(0+y)-f(0))/y
= f(x) f'(0)
= f(x) #