 Generalized Leibniz rule

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.