Algebra extensions and derived-discrete algebras

Let $\phi\colon A\rightarrow B$ be an algebra extension. We prove that if $\phi$ is split, the derived-discreteness of $A$ implies the derived-discreteness of $B$; if $\phi$ is separable and the right $A$-module $B$ is projective, the converse holds. We prove an analogous statement for piecewise hereditary algebras.