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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

This can be proved 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.

ProofEdit

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 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

ReferencesEdit

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