{{dablink|For Euler's chain rule relating partial derivatives of three independent variables, see
Triple product ruleThe triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables...
.}}
{{dablink|For the counting principle in combinatorics, see
Rule of productIn combinatorics, the rule of product or multiplication principle is a basic counting principle...
.}}
{{Calculus|cTopic=Differentiation}}
In
calculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, the
product rule is a formula used to find the
derivativeIn 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 one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s of products of two or more functions. It may be stated thus:
or in the
Leibniz notationIn calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent "infinitely small" increments of x and y, just as Δx and Δy represent finite increments of x and y...
thus:
.
The derivative of the product of three functions is:
.
Discovery by Leibniz
Discovery of this rule is credited to
Gottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
(however, Child (2008) argues that it is due to
Isaac BarrowIsaac Barrow was an English Christian theologian, and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was...
), who demonstrated it using
differentialsIn 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 Δx . The differential dx represents such a change, but is infinitely small...
. Here is Leibniz's argument: Let
u(
x) and
v(
x) be two
differentiable functionIn calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...
s of
x. Then the differential of
uv is
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\begin{align}
y + d(u\cdot v) & {} = (u + du)\cdot (v + dv) \\
d(u\cdot v) & {} = (u + du)\cdot (v + dv) - u\cdot v \\
& {} = u\cdot dv + v\cdot du + du\cdot dv.
\end{align}
Since the term
du·
dv is "negligible" (compared to
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
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NEWLINE- Suppose we want to differentiate ƒ(x) = x2 sin
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
(x). By using the product rule, one gets the derivative ƒ '(x) = 2x sin(x) + x2cos(x) (since the derivative of x2 is 2x and the derivative of sin(x) is cos(x)). NEWLINE- One special case of the product rule is the constant multiple rule which states: if c is a real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
and ƒ(x) is a differentiable function, then cƒ(x) is also differentiable, and its derivative is (c × ƒ)'(x) = c × ƒ '(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 linearIn mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
. NEWLINE- The rule for 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 integrals...
is derived from the product rule, as is (a weak version of) the quotient rule. (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.)
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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, there are clear
counterexampleIn logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
s. Consider a differentiable function
ƒ(
x) whose derivative is
ƒ '(
x). This function can also be written as
ƒ(
x) · 1, since 1 is the
identity elementIn 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. If the above-mentioned misconception were true, (
u′)(
v′) would equal
zero0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
. This is true because the
derivative of a constantIn calculus, the derivative of a constant function is zero .The rule can be justified in various ways...
(such as 1) is zero and the product of
ƒ '(
x) · 0 is also zero.
Proof of the product rule
A rigorous proof of the product rule can be given using the properties of
limitsIn mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
and the definition of the derivative as a limit of
NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
's
difference quotientThe primary vehicle of calculus and other higher mathematics is the function. Its "input value" is its argument, usually a point expressible on a graph...
.
If
and
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.
The region between the smaller and larger rectangle can be split into two rectangles, the sum of whose areas is
Therefore the expression in (1) is equal to
Assuming that all limits used exist, (4) is equal to
Now
This holds because
f(
x) remains constant as
w →
x.
This holds because differentiable functions are continuous (
g is assumed differentiable in the statement of the product rule).
Also:
and
because
f and
g are differentiable at
x;
We conclude that the expression in (5) is equal to
A Brief Proof
By definition, if
are differentiable at
then we can write
such that
, that is,
. Then:
Taking the limit for small
gives the result.
Using logarithms
Let
f =
uv and suppose
u and
v are positive functions of
x. Then
Differentiating both sides:
and so, multiplying the left side by
f, and the right side by
uv,
The proof appears in
http://planetmath.org/encyclopedia/LogarithmicProofOfProductRule.html. Note that since
u,
v need to be continuous, the assumption on positivity does not diminish the generality.
This proof relies on the
chain ruleIn calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
and on the properties of the
natural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational and transcendental 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.
Using the chain rule
The product rule can be considered a special case of the
chain ruleIn calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
for several variables.
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Using non-standard analysis
Let
u and
v be continuous functions in
x, and let d
x, d
u and d
v be
infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
s within the framework of
non-standard analysisNon-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
, specifically the
hyperreal numberThe system of hyperreal numbers represents a rigorous method of treating the infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form1 + 1 + \cdots + 1. \, Such a number is...
s. Using st to denote the
standard part functionIn non-standard analysis, the standard part function is a function from the limited hyperreals to the reals, which associates to every hyperreal, the unique real infinitely close to it. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de...
that associates to a finite hyperreal number the real infinitely close to it, this gives
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Using smooth infinitesimal analysis
In the context of Lawvere's approach to infinitesimals, let du and dv be nilsquare infinitesimals. Then
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\begin{align}
d(uv) & {} = (u + du)(v + dv) -uv \\
& {} = uv + u\cdot dv + v\cdot du + du\cdot dv - uv \\
& {} = u\cdot dv + v\cdot du + du\cdot dv \\
& {} = u\cdot dv + v\cdot du\,\!
\end{align}
provided that
(this may not actually be true even for nilsquare infinitesimals in general).
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