Given a rational map $f:\hat{\mathbb{C}} \to \hat{\mathbb{C}}$, where $\hat{\mathbb{C}}$ is the Riemann sphere, I need to prove that $2\deg(f) - 2 = \sum (v_f(p)-1)$, i.e. prove the Riemann-Hurwitz formula, but without using that, just using the definition of the multiplicity on a point.
I already know when $f(z)=p(z)/q(z)$ with $\deg(f)=\deg p(z) = \deg q(z)$, but I don't know how to do it on the general case when the degree of the polynomials are different.