= Generalized Leibniz rule
{title2=$\displaystyle D^n\prod_{j=1}^m f_j=\sum_{k_1+\cdots+k_m=n}\frac{n!}{k_1!\cdots k_m!}\prod_{j=1}^mD^{k_j}f_j$}
For $m$ functions with sufficiently many <derivatives>, the $n$th <derivative> of their product distributes the $n$ differentiations among all factors. If factor $j$ is differentiated $k_j$ times, its term has the <multinomial coefficient> $n!/(k_1!\cdots k_m!)$, where the nonnegative integers $k_j$ sum to $n$. Induction on the number of factors proves this from the two-factor <Leibniz rule>: splitting off the final factor gives a <binomial coefficient>, which multiplies the previous multinomial coefficient to give the one displayed above. Equivalently, count the ways to choose which of the $n$ successive applications of the <product rule> differentiate each factor.
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