Real exponential function/Characterization with derivative/Exercise

Let

$f\colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto f(x),$

be a differentiable function with the property

$f'=f{\text{ and }}f(0)=1.$

Prove that ${}f(x)=\exp x$ for all ${}a\in \mathbb {R}$.

Create a solution