Question

f(x+y)=f(x)f(y) 且f'(0)=1 求f'(x)

Answer 1

简单分析一下，答案如图所示

Answer 2

解：∵f(x+y)=f(x)f(y)
∴令y=0，得f(x)=f(x)f(0)==>f(x)(f(0)-1)=0.........(1)
∵f'(0)=1............(2)
∴f(x)≠0
∴由(1)得f(0)-1=0
==>f(0)=1..........(3)
故f'(x)=lim(h->0)[(f(x+h)-f(x))/h]
(由导数定义得)
=lim(h->0)[(f(x)f(h)-f(x))/h]
=lim(h->0)[f(x)(f(h)-1)/h]
=f(x)lim(h->0)[(f(h)-1)/h]
=f(x)lim(h->0)[(f(h)-f(0))/h]
(由(3)式得)
=f(x)f'(0)
(由导数定义得)
=f(x)*1
(由(2)式得)
=f(x)