"Let E be a function satisfying E(0)=E'(0)=1. Prove that if E(a+b)=E(a)*E(b) for all a and b then E is differentiable and and E'(x)=E(x) for all x. Find an example of a function satisfying E(a+b)=E(a)*E(b)."
I did a pretty thorough search on this site to see if there was anything similar to this problem, but the only ones were trying to prove continuity using the epsilon>0 and delta>0 definitions, and perhaps that has something to do with finding the answer, but I'm not smart enough to figure out how it might. Also, the professor gave a hint that I need use the limit definition of a derivative to prove that E is differentiable, if that's relevant. Any help is greatly appreciated!
$lim_{h\rightarrow 0}{{E(x+h)-E(x)}\over h}=lim_{h\rightarrow 0}{{E(x)E(h)-E(x)}\over h}$
$lim_{h\rightarrow 0}E(x){{E(h)-1}\over h}=E(x)E'(0)=E(x)$.
Take any $a>0$ and $E(x)=a^x$.