General Leibniz rule

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

In calculus, the general 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 binomial coefficient.

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

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

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

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:

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

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.

See also

• Derivation (abstract algebra)
• Differential algebra
• Product rule
• Derivative

External links

• Planet Math