General Leibniz rule


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 denotes the jth derivative of f (and in particular ).

The rule can be proven by using the product rule and mathematical induction.

Second derivativeEdit

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


More than two factorsEdit

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


where the sum extends over all m-tuples (k1,...,km) of non-negative integers with   and


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


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




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

Multivariable calculusEdit

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 alsoEdit


  1. ^ Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319. ISBN 9780387950006.