Product rule - Citizendia

In calculus, the product rule also called Leibniz's law (see derivation), governs the differentiation of products of differentiable functions. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. In Mathematics, the limit of a function is a fundamental concept in Calculus and analysis concerning the behavior of that function near a particular In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner In Mathematics, matrix calculus is a specialized notation for doing Multivariable calculus, especially over spaces of matrices, where it defines the In Calculus, the mean value theorem states roughly that given a section of a smooth curve there is at least one point on that section at which the Derivative In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change 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 In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. In Mathematics, an implicit function is a generalization for the concept of a function in which the Dependent variable has not been given "explicitly" In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor In Differential calculus, related rates problems involve finding a rate that a quantity changes by relating that quantity to other quantities whose rates of change The primary operation in Differential calculus is finding a Derivative. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space See the following pages for lists of Integrals: List of integrals of rational functions List of integrals of irrational functions In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other Disk integration is a means of calculating the Volume of a Solid of revolution, when integrating along the axis of revolution Shell integration (the shell method in Integral calculus) is a means of calculating the Volume of a Solid of revolution, when integrating In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires In Mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions In Integral calculus, the use of Partial fractions is required to integrate the general Rational function. In Calculus, interchange of the order of integration is a methodology that transforms multiple integrations of functions into other hopefully simpler integrals by Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

It may be stated thus:

$(fg)'=f'g+fg' \,$

or in the Leibniz notation thus:

${d\over dx}(uv)=u{dv\over dx}+v{du\over dx}.$

1 Discovery by Leibniz
2 Examples
3 A common error
4 Proof of the product rule
5 Alternative proof: using logarithms
6 Alternative proof: using the chain rule
7 Generalizations
8 An application
9 See also

Discovery by Leibniz

Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. In Calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz In mathematics and more specifically in Differential calculus, the term differential has several interrelated meanings Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x. Then the differential of uv is

$\begin{align} d(uv) & {} = (u + du)(v + dv) - uv \\ & {} = u\,dv + v\,du + du\,dv. \end{align}$

Since the term (du)(dv) is "negligible" (i. e. at least quadratic in du and dv), Leibniz concluded that

$d(uv) = v\,du + u\,dv \,$

and this is indeed the differential form of the product rule. In mathematics the term quadratic describes something that pertains to squares, to the operation of Squaring, to terms of the second degree, or equations If we divide through by the differential dx, we obtain

$\frac{d}{dx} (uv) = v \left( \frac{du}{dx} \right) + u \left( \frac{dv}{dx} \right)$

which can also be written in "prime notation" as

$(uv)' = v u' + u v'. \,$

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). In Mathematics, the real numbers may be described informally in several different ways 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. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that
The product rule can be used to derive the rule for integration by parts and (weak version of) the quotient rule. In Calculus, and more generally in Mathematical analysis, integration by parts is a rule that transforms the Integral of products of functions into other 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 (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 counterexamples to this. In Logic, and especially in its applications to Mathematics and Philosophy, a counterexample is an exception to a proposed general rule i 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. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that 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 Calculus, the Derivative of a Constant function is zero (A constant function is one that does not depend on the independent variable such

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. In Mathematics, the concept of a " limit " is used to describe the Behavior of a function as its argument either "gets close" Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements The primary vehicle of Calculus and other higher mathematics is the function.

Suppose

$h(x) = f(x)g(x),\,$

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

$h'(x) = \lim_{w\to x}{ h(w) - h(x) \over w - x} = \lim_{w\to x}{f(w)g(w) - f(x)g(x) \over w - x}. \qquad\qquad(1)$

Now the difference

$f(w)g(w) - f(x)g(x)\qquad\qquad(2)$

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

$f(x) \Bigg( g(w) - g(x) \Bigg) + g(w)\Bigg( f(w) - f(x) \Bigg).\qquad\qquad(3)$

(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

$\lim_{w\to x}\left( f(x) \left( {g(w) - g(x) \over w - x} \right) + g(w)\left( {f(w) - f(x) \over w - x} \right) \right).\qquad\qquad(4)$

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

$\left(\lim_{w\to x}f(x)\right) \left(\lim_{w\to x} {g(w) - g(x) \over w - x}\right) + \left(\lim_{w\to x} g(w)\right) \left(\lim_{w\to x} {f(w) - f(x) \over w - x} \right). \qquad\qquad(5)$

Now

$\lim_{w\to x}f(x) = f(x)\,$

because f(x) remains constant as w → x;

$\lim_{w\to x} {g(w) - g(x) \over w - x} = g'(x)$

because g is differentiable at x;

$\lim_{w\to x} {f(w) - f(x) \over w - x} = f'(x)$

because f is differentiable at x;

and now the "hard" one:

$\lim_{w\to x} g(w) = g(x)\,$

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

$f(x)g'(x) + g(x)f'(x). \,$

Alternative proof: using logarithms

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

$\ln f = \ln u + \ln v.\,$

Differentiating both sides:

${1 \over f} {d \over dx} f = {1 \over u} {d \over dx} u + {1 \over v} {d \over dx} v$

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

${d \over dx} f = v {d \over dx} u + u {d \over dx} v.$

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. 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 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.

${d (ab) \over dx} = \frac{\partial(ab)}{\partial a}\frac{da}{dx}+\frac{\partial (ab)}{\partial b}\frac{db}{dx} = b \frac{da}{dx} + a \frac{db}{dx}.$

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

$\frac{d(uvw)}{dx} = \frac{du}{dx}vw + u\frac{dv}{dx}w + uv\frac{dw}{dx}.$

For a collection of functions $f_1, \dots, f_k$ , we have

$\frac{d}{dx} \prod_{i=1}^k f_i(x) = \left(\sum_{i=1}^k \frac{\frac{d}{dx} f_i(x)}{f_i(x)}\right) \prod_{i=1}^k f_i(x).$

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

$y^{(n)}(x) = \sum_{k=0}^n {n \choose k} u^{(n-k)}(x)\; v^{(k)}(x).$

See also binomial coefficient and the formally quite similar binomial theorem. In Calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the Product rule. In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x k term in the Polynomial In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says See also Leibniz rule (generalized product rule). In Calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the Product rule.

Higher partial derivatives

For partial derivatives, we have

${\partial^n \over \partial x_1\,\cdots\,\partial x_n} (uv) = \sum_S {\partial^{|S|} u \over \prod_{i\in S} \partial x_i} \cdot {\partial^{n-|S|} v \over \prod_{i\not\in S} \partial x_i}$

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:

$\begin{align} &{}\quad {\partial^3 \over \partial x_1\,\partial x_2\,\partial x_3} (uv) \\ \\ &{}= u \cdot{\partial^3 v \over \partial x_1\,\partial x_2\,\partial x_3} + {\partial u \over \partial x_1}\cdot{\partial^2 v \over \partial x_2\,\partial x_3} + {\partial u \over \partial x_2}\cdot{\partial^2 v \over \partial x_1\,\partial x_3} + {\partial u \over \partial x_3}\cdot{\partial^2 v \over \partial x_1\,\partial x_2} \\ \\ &{}\qquad + {\partial^2 u \over \partial x_1\,\partial x_2}\cdot{\partial v \over \partial x_3} + {\partial^2 u \over \partial x_1\,\partial x_3}\cdot{\partial v \over \partial x_2} + {\partial^2 u \over \partial x_2\,\partial x_3}\cdot{\partial v \over \partial x_1} + {\partial^3 u \over \partial x_1\,\partial x_2\,\partial x_3}\cdot v. \end{align}$

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. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a bilinear map is a function of two arguments that is linear in each 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

$(D_\left( x,y \right)\,B)\left( u,v \right) = B\left( u,y \right) + B\left( x,v \right)\qquad\forall (u,v)\in X \times Y.$

Derivations in abstract algebra

In abstract algebra, the product rule is used to define what is called a derivation, not vice versa. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Abstract algebra, a derivation is a function on an algebra which generalizes certain features of the Derivative operator

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). In mathematics anticommutativity refers to the property of an operation being anticommutative, i Here, the product rule must be calculated as

$(fg)'=f'g+fg' \,$

and not

$(fg)'=f'g+g'f \,$ , even though this would be correct for multiplication of scalars.

An application

Among the applications of the product rule is a proof that

${d \over dx} x^n = nx^{n-1}$

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. 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 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

$\begin{align} {d \over dx}x^{n+1} &{}= {d \over dx}\left( x^n\cdot x\right) \\ \\ &{}= x{d \over dx} x^n + x^n{d \over dx}x \qquad\mbox{(the product rule is used here)} \\ \\ &{}= x\left(nx^{n-1}\right) + x^n\cdot 1\qquad\mbox{(the induction hypothesis is used here)} \\ \\ &{}= (n + 1)x^n. \end{align}$

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

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