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 (see proof below).

Second derivative

In case :

The binomial coefficients can be deduced thanks to the Pascal's triangle.

More than two factors

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

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

References

  1. ^ Olver, Applications of Lie groups to differential equations, page 318
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