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In calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
, the product rule (also called Leibniz's law; see derivation
Derivation (abstract algebra)

In abstract algebra, a derivation is a function on an algebra over a field which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field F, an F-derivation is an F-linear map D: A → A that satisfies Product rule:...
) is a formula used to find the derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
s of products of functions. It may be stated thus:

or in the Leibniz notation
Leibniz notation

In calculus, Leibniz's notation, named in honor of the 17th-century Germany philosophy and mathematics Gottfried Leibniz, was originally the use of expressions such as dx and dy and to represent "infinitely small" increments of quantities x and y, just as ?x and ?y represent finite increments of x and y respe...
 thus:
Discovery by Leibniz
Discovery of this rule is credited to Gottfried Leibniz
Gottfried Leibniz

Gottfried Wilhelm Leibniz was a Germany polymath who wrote primarily in Latin and French language.He occupies an equally grand place in both the history of philosophy and the history of mathematics....
, who demonstrated it using differentials
Differential (calculus)

In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable, then a change in the value of x is often denoted ?x ....
. Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x.






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In calculus
Calculus

Calculus is a branch of mathematics that includes the study of limit , derivatives, integrals, and infinite series, and constitutes a major part of modern university education....
, the product rule (also called Leibniz's law; see derivation
Derivation (abstract algebra)

In abstract algebra, a derivation is a function on an algebra over a field which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field F, an F-derivation is an F-linear map D: A → A that satisfies Product rule:...
) is a formula used to find the derivative
Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
s of products of functions. It may be stated thus:

or in the Leibniz notation
Leibniz notation

In calculus, Leibniz's notation, named in honor of the 17th-century Germany philosophy and mathematics Gottfried Leibniz, was originally the use of expressions such as dx and dy and to represent "infinitely small" increments of quantities x and y, just as ?x and ?y represent finite increments of x and y respe...
 thus:

Discovery by Leibniz


Discovery of this rule is credited to Gottfried Leibniz
Gottfried Leibniz

Gottfried Wilhelm Leibniz was a Germany polymath who wrote primarily in Latin and French language.He occupies an equally grand place in both the history of philosophy and the history of mathematics....
, who demonstrated it using differentials
Differential (calculus)

In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable, then a change in the value of x is often denoted ?x ....
. Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x. Then the differential of uv is



Since the term du·dv is "negligible" (i.e. at least quadratic
Quadratic

In mathematics, the term quadratic describes something that pertains to Square , to the operation of squaring, to terms of the second degree of a polynomial, or equations or formulas that involve such terms....
 in du and dv), 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 "prime notation" as

Examples


  • Suppose one wants to differentiate f(x) = x2 sin
    Siné

    Maurice Sinet, known as Sin? is a France cartoonist.As a young man he studied drawing and graphic arts, earning his life as a cabaret singer....
    (x). By using the product rule, you get the derivative f'(x) = 2x sin(x) + x2cos(x) (since the derivative of x2 is 2x and the derivative of sin(x) is cos(x)).
  • One special case of the product rule is the constant multiple rule which states: if c is a real number
    Real number

    In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
     and f(x) is a differentiable function, then cf(x) is also differentiable, and its derivative is (c × f)'(x) = c × f '(x). This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear
    Linear transformation

    In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication....
    .
  • The product rule can be used to derive the rule for integration by parts
    Integration by parts

    In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals....
     and (weak version of) the quotient rule
    Quotient rule

    In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist....
    . (It is a "weak" version in that it does not prove that the quotient is differentiable, but only says what its derivative is if it is differentiable.)


A common error


It is a common error, when studying calculus, to suppose that the derivative of (uv) equals (u′)(v′) (Leibniz himself made this error initially); however, it is quite easy to find counterexample
Counterexample

In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i.e., a specific instance of the falsity of a universal quantification ....
s to this. Most simply, take a function f(x), whose derivative is f '(x). Now that function can also be written as f(x) · 1, since 1 is the identity element
Identity element

In mathematics, an identity element is a special type of element of a Set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them....
 for multiplication. Suppose the above-mentioned misconception were true; if so, (u′)(v′) would equal zero
0 (number)

0 is both a number and the numerical digit used to represent that number in numeral system. It plays a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures....
. This is true because the derivative of a constant
Derivative of a constant

In calculus, the derivative of a constant function is 0 .The rule can be justified in various ways. The derivative is the slope of the tangent to the given function's graph, and the graph of a constant function is a horizontal line, whose slope is zero....
 (such as 1) is zero and the product of f '(x) · 0 is also zero.

Proof of the product rule


A rigorous proof of the product rule can be given using the properties of limits
Limit (mathematics)

In mathematics, the concept of a "limit" is used to describe the behavior of a Function as its argument or input either "gets close" to some point, or as the argument becomes arbitrarily large; or the behavior of a sequence's elements as their index increases indefinitely....
 and the definition of the derivative as a limit of Newton
Isaac Newton

Sir Isaac Newton, Fellow of the Royal Society was an English people physicist, mathematician, Astronomy, Natural philosophy, Alchemy, and Theology and one of the the 100 in human history....
's difference quotient
Difference quotient

The primary vehicle of calculus and other higher mathematics is the Function . Its "input value" is its argument, usually a point expressible on a graph....
.

Suppose

and that ƒ and g are each differentiable at the fixed number x. Then

Now the difference

is the area of the big rectangle minus the area of the small rectangle in the illustration. That L-shaped region can be split into two rectangles, the sum of whose areas is readily seen to be

(The illustration disagrees with some special cases, since ƒ(w) need not actually be bigger than ƒ(x) and g(w) need not actually be bigger than g(x). Nonetheless, the equality of (2) and (3) is easily checked by algebra.)

Therefore the expression in (1) is equal to

If all four of the limits in (5) below exist, then the expression in (4) is equal to

Now

because ƒ(x) remains constant as w ? x;

because g is differentiable at x;

because ƒ is differentiable at x;

and now the "hard" one:

because g, being differentiable, is continuous at x.

We conclude that the expression in (5) is equal to

Alternative proof


This proof is similar to the proof above. Suppose

By applying Newton's difference quotient and the limit as h approaches 0, we are able to represent the derivative in the form

In order to simplify this limit we add and subtract the term to the numerator, keeping the fraction's value unchanged

This allows us to factorise the numerator like so

The fraction is split into two

The limit is applied to each term and factor of the limit expression

Each limit is evaluated. Taking into consideration the definition of the derivative, the result is

Alternative proof: using logarithms

Let f = uv and suppose u and v are positive. Then

Differentiating both sides:

and so, multiplying the left side by f, and the right side by uv,

The proof appears in . Note that since u, v need to be continuous, the assumption on positivity does not diminish the generality.

This proof relies on the chain rule
Chain rule

In calculus, the chain rule is a formula for the derivative of the functional composition of two function .In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of Mathematics#Change of y with respect to x can be computation as the rate of chan...
 and on the properties of the natural logarithm
Natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e , where e is an irrational number constant approximately equal to 2.718281828....
 function, both of which are deeper than the product rule. From one point of view, that is a disadvantage of this proof. On the other hand, the simplicity of the algebra in this proof perhaps makes it easier to understand than a proof using the definition of differentiation directly.

Alternative proof: using the chain rule


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



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 Leibniz rule
Leibniz rule (generalized product rule)

In calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. It states that if f and g are n-times differentiable functions, then the nth derivative of the product fg is given by...
 for higher derivatives of a product of two factors:

See also binomial coefficient
Binomial coefficient

In mathematics, the binomial coefficient is the coefficient of the x k term in the polynomial expansion of the binomial exponentiation  n....
 and the formally quite similar binomial theorem
Binomial theorem

In mathematics, the binomial theorem is an important formula giving the expansion of exponentiation of sums. Its simplest version states that...
. See also Leibniz rule (generalized product rule)
Leibniz rule (generalized product rule)

In calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. It states that if f and g are n-times differentiable functions, then the nth derivative of the product fg is given by...
.

Higher partial derivatives


For partial derivatives, we have

where the index S runs through the whole list of 2n subsets of . If this seems hard to understand, consider the case in which n = 3:

A product rule in Banach spaces


Suppose X, Y, and Z are Banach space
Banach space

In mathematics, Banach spaces are one of the central objects of study in functional analysis. They are topological vector spaces that have many interesting properties associated with them....
s (which includes Euclidean space
Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
) and B : X × Y ? Z is a continuous
Continuous function (topology)

In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f always contain the of a set of points near x....
 bilinear operator
Bilinear operator

In mathematics, a bilinear map is a function of two arguments that is linear map in each. An example of such a map is multiplication of integers....
. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D(x,y)B : X × Y ? Z given by

Derivations in abstract algebra


In abstract algebra
Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as group , ring , field , module , vector spaces, and algebra over a field....
, the product rule is used to define what is called a derivation
Derivation (abstract algebra)

In abstract algebra, a derivation is a function on an algebra over a field which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field F, an F-derivation is an F-linear map D: A → A that satisfies Product rule:...
, not vice versa.

For vector functions


For the product rule regarding vector functions, where the result of the function is a vector, the product rule changes somewhat due to the anticommutative
Anticommutativity

In mathematics, anticommutativity refers to the property of an Operation being anticommutative, i.e. being non-Commutativity in a precise way....
 properties of vector products (multiplying vectors and getting a vector as a product). Here, the product rule must be calculated as and not , even though this would be correct for multiplication of scalars.

An application


Among the applications of the product rule is a proof that

when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction
Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then...
 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 of n, it is true also of n + 1.

See also

  • Quotient rule
    Quotient rule

    In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist....
  • Reciprocal rule
    Reciprocal rule

    In calculus, the reciprocal rule is a shorthand method of finding the derivative of a function that is the Multiplicative inverse of a differentiable function, without using the quotient rule or chain rule....
  • Chain rule
    Chain rule

    In calculus, the chain rule is a formula for the derivative of the functional composition of two function .In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of Mathematics#Change of y with respect to x can be computation as the rate of chan...
  • Integration by parts
    Integration by parts

    In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, hopefully simpler, integrals....
  • Differential (calculus)
    Differential (calculus)

    In calculus, a differential is traditionally an infinitesimally small change in a variable. For example, if x is a variable, then a change in the value of x is often denoted ?x ....
  • Derivation (abstract algebra)
    Derivation (abstract algebra)

    In abstract algebra, a derivation is a function on an algebra over a field which generalizes certain features of the derivative operator. Specifically, given an algebra A over a ring or a field F, an F-derivation is an F-linear map D: A → A that satisfies Product rule:...