Let $\ f: \mathbb{R} \rightarrow \mathbb{R}\ $ ,has derivative at $0$ and the function is not equal identity to $0$.

In addition for $\forall a,b \in \mathbb{R}$ we have $f(a+b)=f(a)f(b)$

Prove the following claims:

$\quad 1)f$ has detrative in all $\mathbb{R}$

$\quad 2)f(x)=e^{\left( (xf'(0) \right)}$

I was able to prove the section $1$, so you can assume it is true.
I was stuck trying to solve the section $2$.

Thank you