General Leibniz rule

For other uses, see Leibniz's rule (disambiguation).

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Specialized

In calculus, the general Leibniz rule,^[1] named after Gottfried Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if u and v are n-times differentiable functions, then product uv is also n-times differentiable and its nth derivative is given by

(uv)^{(n)}=\sum _{k=0}^{n}{n \choose k}u^{(k)}v^{(n-k)}

where ${n \choose k}={n! \over k!(n-k)!}$ is the binomial coefficient.

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

Second derivative

In case $n=2$ :

(f\cdot g)''=\sum \limits _{k=0}^{2}{C_{k}^{2}f^{(2-k)}g^{(k)}}={f''g}+{2f'g'}+{fg''}.

More than two factors

The formula can be generalized to the product of m differentiable functions f₁,...,f_m.

\left(f_{1}f_{2}\cdots f_{m}\right)^{(n)}=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}{n \choose k_{1},k_{2},\ldots ,k_{m}}\prod _{1\leq t\leq m}f_{t}^{(k_{t})}\,,

where the sum extends over all m-tuples (k₁,...,k_m) of non-negative integers with $\sum _{t=1}^{m}k_{t}=n$ and

{n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots k_{m}!}}

are the multinomial coefficients. This is akin to the multinomial formula from algebra.

Multivariable calculus

With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally:

\partial ^{\alpha }(fg)=\sum _{\{\beta \,:\,\beta \leq \alpha \}}{\alpha \choose \beta }(\partial ^{\beta }f)(\partial ^{\alpha -\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.

Notes

↑ Olver, Applications of Lie groups to differential equations, page 318

External links

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