General Leibniz rule
In calculus, the general Leibniz rule, 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
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where is the binomial coefficient and
This can be proved by using the product rule and mathematical induction.
If, for example, n = 2, the rule gives an expression for the second derivative of a product of two functions:
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
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.
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.
- Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319.