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
where is the binomial coefficient and .
This can be proved by using the product rule and mathematical induction (see proof below).
In case :
The binomial coefficients can be deduced thanks to the Pascal's triangle.
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
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, Applications of Lie groups to differential equations, page 318