Stability of Tschirnhausen Bundles

Let $\alpha :X\to Y$ be a general degree $r$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $r$. We prove that the Tschirnhausen bundle of $\alpha$ is semistable if $g(Y)\ge 1$ and stable if $g(Y)\ge 2$.