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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 n-times differentiable functions, then the product is also n-times differentiable and its n-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 derivative

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

More than two factors

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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

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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

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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

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References

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