Let $C/K$ be a smooth curve and $f \in K(C)$ be a function. Then by identifying $f$ with a rational map, we can get a 1-1 correspondence with maps $C \to \mathbf{P}^1$, with one direction being given by $P \mapsto [f(P),1]$.
In the situation as above, $$ \textrm{div} (f) = f^* ((0) - (\infty))$$
This is supposed to follow directly from the definitions. On the one hand, I know that
$$ \textrm{div} (f) = \sum_{P \in C} v_P (f) (P) $$
For the right hand side,
$$ f^* ((0) - (\infty)) = f^* ((0)) - f^* (( \infty ))$$ At this point, I can intuitively see the result: we count the zeros with multiplicities and the poles with multiplicities and those are all the nonzero divisors of $f$. However, I would like to see a rigorous proof involving the definitions, valuations, etc.