General Leibniz rule

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

In calculus, the general Leibniz rule, 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