General Leibniz rule

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.

More than two factors

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

$\left(f_1 f_2 \cdots f_m\right)^{(n)}=\sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m} \prod_{1\le t\le m}f_{t}^{(k_{t})}\,,$

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:

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

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.

See also

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

Notes

1. Olver, Applications of Lie groups to differential equations, page 318

External links