It transforms products into sums

Calculus Level 5

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

The function f:R+Rf : \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,yx,y. In addition, ff is continuous at x=1x=1, and f(3)=7f(3) = 7.

Determine whether ff is necessarily differentiable at 11. If it is, compute f(1)f'(1). If it isn't, simply input 00 as your answer.

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