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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