# General Leibniz rule

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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 *f* and *g* are *n*-times differentiable functions, then product *fg* is also *n*-times differentiable and its *n*th derivative is given by

where 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 *f*_{1},...,*f*_{m}.

where the sum extends over all *m*-tuples (*k*_{1},...,*k*_{m}) of non-negative integers with and

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:

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 . Since *R* is also a differential operator, the symbol of *R* is given by:

A direct computation now gives:

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

## Notes

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