General Leibniz rule
Part of a series of articles about  
Calculus  





Specialized


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).
Contents
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 f_{1},...,f_{m}.
where the sum extends over all mtuples (k_{1},...,k_{m}) of nonnegative integers with and
are the multinomial coefficients. This is akin to the multinomial formula from algebra.
Proof
Show that the equality holds for any functions and that are times differentiable functions.
Basis At rank we get:
and by the product rule.
Hence, the equality holds at the initial rank.
Inductive step We assume that the equality
holds for .
Therefore, at rank we get:
Multivariable calculus
With the multiindex 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
 ^ Olver, Applications of Lie groups to differential equations, page 318