In calculus, the product rule also called Leibniz's law (see derivation), governs the differentiation of products of differentiable functions. For other uses, see Calculus (disambiguation). ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ... For other uses, see Derivative (disambiguation). ... This article is about functions in mathematics. ...

It may be stated thus:

or in the Leibniz notation thus: In calculus, the Leibniz notation, named in honor of the 17th century German philosopher and mathematician Gottfried Wilhelm Leibniz (pronounced LIBE nits) was originally the use of dx and dy and so forth to represent infinitely small increments of quantities x and y, just as Δx and Δy represent finite...

Discovery by Leibniz

Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x. Then the differential of uv is Leibniz redirects here. ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ...

Since the term (du)(dv) is "negligible" (i.e. at least quadratic in du and dv), Leibniz concluded that f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where a is nonzero. ...

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(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 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.
• The product rule can be used to derive the rule for integration by parts and (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.)

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... 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. ... 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. ...

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 counterexamples 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 for multiplication. Suppose the above-mentioned misconception were true; if so, (u′)(v′) would equal zero. This is true because the derivative of a constant (such as 1) is zero and the product of f '(x) · 0 is also zero. In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i. ... For other uses, see identity (disambiguation). ... Zero redirects here. ... In calculus, the derivative of a constant function is zero. ...

Proof of the product rule

A rigorous proof of the product rule can be given using the properties of limits and the definition of the derivative as a limit of Newton's difference quotient. Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... The primary vehicle of calculus and other higher mathematics is the function. ...

Suppose

and that f 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 Image File history File links Size of this preview: 800 × 320 pixelsFull resolution (1257 × 503 pixel, file size: 9 KB, MIME type: image/png) Illustration to accompany a proof of the product rule. ...

(The illustration disagrees with some special cases, since f(w) need not actually be bigger than f(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 f(x) remains constant as w → x;

because g is differentiable at x;

because f is differentiable at x;

and now the "hard" one:

because g is continuous at x. How do we know g is continuous at x? Because another theorem says differentiable functions are continuous.

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

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 [1]. 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 and on the properties of the natural logarithm 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. In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ...

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 for higher derivatives of a product of two factors: if y = uv and y⁽ⁿ⁾ denotes the n-th derivative of y, then In calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. ...

See also binomial coefficient and the formally quite similar binomial theorem. See also Leibniz rule (generalized product rule). In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... In calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. ...

Higher partial derivatives

For partial derivatives, we have

where the index S runs through the whole list of 2ⁿ subsets of {1, ..., n}. If this seems hard to understand, consider the case in which n = 3:

A product rule in Banach spaces

If X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y → Z is a continuous bilinear operator. 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 In mathematics, Banach spaces (pronounced ), named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

Derivations in abstract algebra

In abstract algebra, the product rule is used to define what is called a derivation, not vice versa.. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

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 properties of vector products (multiplying vectors and getting a vector as a product). Here, the product rule must be calculated as A mathematical operator (typically a binary operator, represented by *) is anticommutative if and only if it is true that x * y = −(y * x) for all x and y on the operators valid domain (e. ...

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 on the exponent n. If n = 0 then xⁿ is constant and nx^n − 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 Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ...

Therefore if the proposition is true of n, it is true also of n + 1.

See also

• Quotient rule
• Reciprocal rule
• Chain rule
• Integration by parts
• Differential (calculus)
• Derivation (abstract algebra)