Leibniz rule

The higher-order product rule is

(f g)^{(n)} = \sum_{k = 0}^{n} (k n) f^{(k)} g^{(n - k)} .

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- Generalized Leibniz rule Leibniz rule

Generalized Leibniz rule ( $D^{n} j = 1 \prod m f_{j} = k_{1} + \dots + k_{m} = n \sum \frac{n !}{k _{1} ! \dots k _{m} !} j = 1 \prod m D^{k_{j}} f_{j}$ )

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}! \dots 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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