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 n-times
differentiable functions, then the product is also n-times differentiable and its n-th derivative is given by
where is the
binomial coefficient and denotes the jth derivative of f (and in particular ).
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
are the
multinomial coefficients. This is akin to the
multinomial formula from algebra.
Proof
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 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.