Composition followed by differentiation between weighted Bergman spaces and weighted Banach spaces of holomorphic functions

Let $\phi$ be an analytic self-map of the open unit disk $\mathbb D$ in the complex plane. Such a map induces through composition a linear composition operator $C_\phi\colon f\mapsto f\circ\phi$. We are interested in the combination of $C_\phi$ weith the differentiation operator $D$, that is in the operator $DC_\phi\colon f\mapsto\phi'\cdot(f\circ\phi)$ acting between weighted Bergman spaces and weighted Banach spaces of holomorphic functions.