Trying to generalize the solving of a functional equation

Question

If we take the following functional equation for $f : \mathbf{R}^+ \rightarrow \mathbf{R}$, such as for any $(x,y), f(xy) = xf(y)+yf(x)$, the solution is trivial if $f$ is derivable (we just derive the equation...). If $f$ is derivable only in $0$, we also can find the solution by considering what happens around $0$. I was wondering if we could do something also if we know that $f$ is derivable in $a, a > 0$. Stuck there. Any help?