A curious multiple derivative identity

Question

Let $z, w$ be two real variables, and $m,n$ two non-negative integers. Is there a simple proof of the following identity? $$\partial_z^{m+1}\partial_w^n\biggl(\frac{z^{n+1}w^{m}}{z-w}\biggr) = (-1)^{m+1}(m+n+1)!\frac{z^{n}w^{m+1}}{(z-w)^{m+n+2}}$$ One fairly cumbersome way in which I managed to prove this involves Taylor expanding $1/(z-w)$ around either $z=0$ or $w=0$, differentiating the series term by term, and resumming. There appears to be something very special about the configuration of powers of $z$ and $w$ that is being differentiated that is leading to this remarkably simple final answer. I would be grateful if anyone could suggest a simpler proof that better explains why this is working.