Leibniz rule (generalized product rule)

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See Leibniz rule, a disambiguation page, for other meanings of this term.

In calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. It states that if f and g are n-times differentiable functions, then the nth derivative of the product fg is given by

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

where {n \choose k} is the usual binomial coefficient.

This can be proved by using the product rule and mathematical induction.

With the multi-index notation the rule states more generally:

\partial^\alpha (fg) = \sum_{\beta \le \alpha} {\alpha \choose \beta} (\partial^{\alpha - \beta} f) (\partial^{\beta} g).

This formula can be used to derive a formula that computes the symbol of the composition of differential operators. In fact, let P and Q be differential operators (with coefficients that are differentiable sufficiently many times.) and R = P \circ Q. Since R is also a differential operator, the symbol of R is given by:

R(x, \xi) = e^{-{\langle x, \xi \rangle}} R (e^{\langle x, \xi \rangle}).

A direct computation now gives:

R(x, \xi) = \sum_\alpha {1 \over \alpha!} {\partial \over \partial \xi}^\alpha P(x, \xi) {\partial \over \partial x}^\alpha Q(x, \xi).

This formula is usually known as the Leibniz formula. It is also used to define the composition in the space of symbols, thereby inducing the ring structure.

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