# It transforms products into sums

Calculus Level 5

$\large f(xy)=f(x)+f(y)$

The function $f : \mathbb{R}^+ \to \mathbb{R}$ with domain the positive real numbers and codomain the real numbers satisfies the above equation for all positive reals $x,y$. In addition, $f$ is continuous at $x=1$, and $f(3) = 7$.

Determine whether $f$ is necessarily differentiable at $1$. If it is, compute $f'(1)$. If it isn't, simply input $0$ as your answer.

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