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

where is the binomial coefficient and

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

## Second derivative[edit]

If, for example, *n* = 2, the rule gives an expression for the second derivative of a product of two functions:

## More than two factors[edit]

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.

## Proof[edit]

The proof of the general Leibniz rule proceeds by induction. Let and be -times differentiable functions. The base case when claims that:

which is the usual product rule and is known to be true. Next, assume that the statement holds for a fixed that is, that

Then,

And so the statement holds for and the proof is complete.

## Multivariable calculus[edit]

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[edit]

- Binomial theorem
- Derivation (differential algebra)
- Derivative
- Differential algebra
- Pascal's triangle

## References[edit]

**^**Olver, Peter J. (2000).*Applications of Lie Groups to Differential Equations*. Springer. pp. 318–319.