Product rule


In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated as
or in Leibniz's notation
The rule may be extended or generalized to many other situations, including to products of multiple functions, to a rule for higher-order derivatives of a product, and to other contexts.

Discovery

Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. Here is Leibniz's argument: Let u and v be two differentiable functions of x. Then the differential of uv is
Since the term du·dv is "negligible", Leibniz concluded that
and this is indeed the differential form of the product rule. If we divide through by the differential dx, we obtain
which can also be written in Lagrange's notation as

Examples

Proof by factoring (from first principles)

Let and suppose that and are each differentiable at. We want to prove that is differentiable at and that its derivative,, is given by. To do this, is added to the numerator to permit its factoring, and then properties of limits are used.
The fact that
is deduced from a theorem that states that differentiable functions are continuous.

Brief proof

By definition, if are differentiable at then we can write
such that also written. Then:
Taking the limit for small gives the result.

Quarter squares

There is a proof using quarter square multiplication which relies on the chain rule and on the properties of the quarter square function :
Differentiating both sides:

Chain rule

The product rule can be considered a special case of the chain rule for several variables.

Non-standard analysis

Let u and v be continuous functions in x, and let dx, du and dv be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers. Using st to denote the standard part function that associates to a hyperreal number the real infinitely close to it, this gives
This was essentially Leibniz's proof exploiting the transcendental law of homogeneity.

Smooth infinitesimal analysis

In the context of Lawvere's approach to infinitesimals, let dx be a nilsquare infinitesimal. Then du = u′ dx and dv = v′ dx, so that
since

Generalizations

A product of more than two factors

The product rule can be generalized to products of more than two factors. For example, for three factors we have
For a collection of functions, we have

Higher derivatives

It can also be generalized to the general Leibniz rule for the nth derivative of a product of two factors, by symbolically expanding according to the binomial theorem:
Applied at a specific point x, the above formula gives:
Furthermore, for the nth derivative of an arbitrary number of factors:

Higher partial derivatives

For partial derivatives, we have
where the index runs through all subsets of, and is the cardinality of. For example, when,

Banach space

Suppose X, Y, and Z are Banach spaces and B : X × Y → Z is a continuous bilinear operator. Then B is differentiable, and its derivative at the point in X × Y is the linear map DB : X × Y → Z given by

Derivations in abstract algebra

In abstract algebra, the product rule is used to define what is called a derivation, not vice versa.

In vector calculus

The product rule extends to scalar multiplication, dot products, and cross products of vector functions, as follows.
For scalar multiplication:
For dot products:
For cross products:
There are also analogues for other analogs of the derivative: if f and g are scalar fields then there is a product rule with the gradient:

Applications

Among the applications of the product rule is a proof that
when n is a positive integer. The proof is by mathematical induction on the exponent n. If n = 0 then xn is constant and nxn − 1 = 0. The rule holds in that case because the derivative of a constant function is 0. If the rule holds for any particular exponent n, then for the next value, n + 1, we have
Therefore, if the proposition is true for n, it is true also for n + 1, and therefore for all natural n.