= Formal pseudodifferential composition rule
{title2=$\partial^n f=\sum_{r\geq0}\binom nr f^{(r)}\partial^{n-r}$}
The generalized <Leibniz rule> extends composition of <differential operators> to all <integer> $n$. Define $\binom nr=n(n-1)\cdots(n-r+1)/r!$. For nonnegative $n$ the sum stops at $r=n$. For $n=-s<0$, its <coefficients> are $(-1)^r\binom{s+r-1}{r}$; in particular $\partial^{-1}f=f\partial^{-1}-f'\partial^{-2}+f''\partial^{-3}-\cdots$. Applying $\partial$ on the left cancels consecutive <derivative> terms and recovers multiplication by $f$, which explains the inverse expansion.
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