General Leibniz rule

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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 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.

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]

References[edit]

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