A function satisfies the equation $f(x+y) = f(x) f(y)$. Show that $f(x)$ is differentiable at all $x$, and determine its $f'(x)$ in terms of $f(x)$.
If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ and $f'(0)$ exists, then $f$ is differentiable on $\mathbb R$