Question

f(x)在R上非0,f(x+y)=f(x)f(y),且f(0)'=1,求f(x)'=f(x)

Answer 1

f(x+0)=f(x)f(0)
∵f(x)≠0,∴f(0)=1
又∵f'(0)=lim(x→0)[f(x)-f(0)]/x=lim(x→0)[f(x)-1]/x=1
∴f'(x)=lim(h→0)[f(x+h)-f(x)]/h
=lim(h→0)[f(x)f(h)-f(x)]/h
=lim(h→0)f(x)[f(h)-1]/h
=f(x)*1
=f(x)

Answer 2

∵f(0+y)=f(0)f(y)
∴f(0)=1
∵f(x+x)=f(x)×f(x)
∴f(x)=a^x
∴f`(x)=lna×f(x)
∵f`(0)=1
∴lna=1
∴a=e
∴f(x)=e^x
∴f`(x)=f(x)