That's an exercise but I'm not so sure how to approach this. Let $k$ be a field of characteristic $p$ and let $f(t)$ be a polynomial in $k[t]$ of degree $d$. Let $C$ be the curve that corresponds to the field extension $k(x,t)$ of $k(t)$, where $x^p + x = f(t)$. Suppose that $C$ is smooth. The problem is then to find the genus of $C$.
The idea is of course to use Hurwitz' theorem but I'm not sure how to rule out wild ramifications. I am also generally interested in how to approach this kind of question (look at valuations, look at the equations in projective space, etc)