In calculus, the general Leibniz rule,^{[1]} named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if $f$ and $g$ are $n$-times differentiable functions, then the product $fg$ is also $n$-times differentiable and its $n$th derivative is given by

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:

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

This page is based on the copyrighted Wikipedia article "General Leibniz rule"; it is used under the Creative Commons
Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may
redistribute it, verbatim or modified, providing that you comply with
the terms of the CC-BY-SA