Real Analysis

Let $D = D_x = \frac{d}{dx}$ be the differentiation-by-$x$ operator on (smooth) functions, so $D f = f'$. Let $\mu = \mu_x$ the multiplication-by-$x$ operator, so $\mu f = x\cdot f$. Did you know the Weyl algebra identity $$(D \mu D)^n = D^n \mu^n D^n?$$ In words, that means applying $D \mu D$ to a (smooth) function $f$ repeatedly $n$ times is the same as applying $D^n$ to $f$ (i.e. applying $D$ repeatedly $n$ times), then $\mu^n$, then $D^n$ again.

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[Source: Communicated by Nikita Lvov in his talk titled "Markov chains in Random Matrix Theory" on February 8, 2018 in the ERGODIC THEORY & STATISTICAL MECHANICS seminar (see abstract here).]