Leibniz rule (generalized product rule)

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 "n"th 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::

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::$R(x,\; xi)\; =\; e^\{-\{langle\; x,\; xi\; angle\; R\; (e^\{langle\; x,\; xi\; angle\}).$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.

External links

* [http://planetmath.org/encyclopedia/GeneralizedLeibnizRule.html Planet Math]