# Differentiability of $f(x+y) = f(x)f(y)$ [duplicate]

Let $f$: $\mathbb R$ $\to$ $\mathbb R$ be a function such that $f(x+y)$ = $f(x)f(y)$ for all $x,y$ $\in$ $\mathbb R$. Suppose that $f'(0)$ exists. Prove that $f$ is a differentiable function.
This is what I've tried: Using the definition of differentiability and taking arbitrary $x_0$ $\in$ $\mathbb R$.