Chain rule - Citizendia

In calculus, the chain rule is a formula for the derivative of the composite of two 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 product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable 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 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 Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a composite function represents the application of one function to the results of another The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the 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 change of y with respect to x can be computed as the rate of change of y with respect to u multiplied by the rate of change of u with respect to x. A variable (ˈvɛərɪəbl is an Attribute of a physical or an abstract System which may change its Value while it is under Observation. Computation is a general term for any type of Information processing.

1 Informal discussion
2 Theorem
3 Examples
4 Chain rule for several variables
5 Proof of the chain rule
6 The fundamental chain rule
7 Tensors and the chain rule
8 Higher derivatives
9 See also
10 References
11 External links

Informal discussion

For an explanation of notation used in this section, see Function composition. In Mathematics, a composite function represents the application of one function to the results of another

The chain rule states that, under appropriate conditions,

$(f \circ g)'(x) = f'(g(x)) g'(x),\,$

which in short form is written as

$(f \circ g)' = f'\circ g\cdot g'$ .

Alternatively, in the Leibniz notation, the chain rule is

$\frac {dy}{dx} = \frac {dy} {du} \frac {du}{dx}.$

In integration, the counterpart to the chain rule is the substitution rule. In Calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires

Theorem

The chain rule in one variable may be stated more completely as follows. ^[1] Let f be a real-valued function on (a,b) which is differentiable at c ∈ (a,b); and g a real-valued function defined on an interval I containing the range of f and f(c) as an interior point. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " If g is differentiable at f(c), then

$(f\circ g)(x)$ is differentiable at x = c, and
$(f\circ g)'(c) = f'(g(c))g'(c).$

Examples

Example I

Suppose that a mountain climber ascends at a rate of 0. 5 kilometers per hour. (For the South African airport with IATA code "KMH" see Johan Pienaar Airport. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 °C per kilometer. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature If one multiplies 6 °C per kilometer by 0. 5 kilometer per hour, one obtains 3 °C per hour. This calculation is a typical chain rule application.

Example II

Consider the function f(x) = (x² + 1)³. Since f(x) = h(g(x)) where g(x) = x² + 1 and h(x) = x³ it follows from the chain rule that

$f '(x) \,$	$= h '(g(x)) g ' (x) \,$	$= 3(g(x))^2(2x) \,$	$= 3(x^2 + 1)^2(2x) \,$
	$= 6x(x^2 + 1)^2. \,$

In order to differentiate the trigonometric function

$f(x) = \sin(x^2),\,$

one can write f(x) = h(g(x)) with h(x) = sin x and g(x) = x². The chain rule then yields

$f'(x) = 2x \cos(x^2) \,$

since h′(g(x)) = cos(x²) and g′(x) = 2x.

Example III

Differentiate arctan(sin x).

$\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$

Thus, by the chain rule,

$\frac{d}{dx}\arctan f(x) = \frac{f'(x)}{1+f^2(x)}\,,$

and in particular,

$\frac{d}{dx}\arctan(\sin x) = \frac{\cos x}{1+\sin^2 x}\,.$

Chain rule for several variables

The chain rule works for functions of more than one variable. Consider the function z = f(x, y) where x = g(t) and y = h(t), and g(t) and h(t) are differentiable with respect to t, then

${\ dz \over dt}={\partial z \over \partial x}{dx \over dt}+{\partial z \over \partial y}{dy \over dt}.$

Suppose that each argument of z = f(u, v) is a two-variable function such that u = h(x, y) and v = g(x, y), and that these functions are all differentiable. Dependent variables and independent variables refer to values that change in relationship to each other Then the chain rule would look like:

${\partial z \over \partial x}={\partial z \over \partial u}{\partial u \over \partial x}+{\partial z \over \partial v}{\partial v \over \partial x}$

${\partial z \over \partial y}={\partial z \over \partial u}{\partial u \over \partial y}+{\partial z \over \partial v}{\partial v \over \partial y}.$

If we considered

$\vec r = (u,v)$

above as a vector function, we can use vector notation to write the above equivalently as the dot product of the gradient of f and a derivative of $\vec r$ :

$\frac{\partial f}{\partial x}=\vec \nabla f \cdot \frac{\partial \vec r}{\partial x}.$

More generally, for functions of vectors to vectors, the chain rule says that the Jacobian matrix of a composite function is the product of the Jacobian matrices of the two functions:

$\frac{\partial(z_1,\ldots,z_m)}{\partial(x_1,\ldots,x_p)} = \frac{\partial(z_1,\ldots,z_m)}{\partial(y_1,\ldots,y_n)} \frac{\partial(y_1,\ldots,y_n)}{\partial(x_1,\ldots,x_p)}.$

Proof of the chain rule

Let f and g be functions and let x be a number such that f is differentiable at g(x) and g is differentiable at x. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. Then by the definition of differentiability,

$g(x+\delta)-g(x)= \delta g'(x) + \epsilon(\delta)\delta \,$

where ε(δ) → 0 as δ → 0. Similarly,

$f(g(x)+\alpha) - f(g(x)) = \alpha f'(g(x)) + \eta(\alpha)\alpha \,$

where η(α) → 0 as α → 0.

Now

$f(g(x+\delta))-f(g(x))\,$	$= f(g(x) + \delta g'(x)+\epsilon(\delta)\delta) - f(g(x)) \,$
	$= \alpha_\delta f'(g(x)) + \eta(\alpha_\delta)\alpha_\delta \,$

where

$\alpha_\delta = \delta g'(x) + \epsilon(\delta)\delta \,$ .

Observe that as δ → 0, α_δ/δ → g′(x) and α_δ → 0, and thus η(α_δ) → 0. It follows that

$\frac{f(g(x+\delta))-f(g(x))}{\delta} \to g'(x)f'(g(x))\mbox{ as } \delta \to 0.$

The fundamental chain rule

The chain rule is a fundamental property of all definitions of derivative and is therefore valid in much more general contexts. For instance, if E, F and G are Banach spaces (which includes Euclidean space) and f : E → F and g : F → G are functions, and if x is an element of E such that f is differentiable at x and g is differentiable at f(x), then the derivative (the Fréchet derivative) of the composition g o f at the point x is given by

$\mbox{D}_x\left(g \circ f\right) = \mbox{D}_{f\left(x\right)}\left(g\right) \circ \mbox{D}_x\left(f\right).$

Note that the derivatives here are linear maps and not numbers. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, the Fréchet derivative is a Derivative defined on Banach spaces Named after Maurice Fréchet, it is commonly used to formalize In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that If the linear maps are represented as matrices (namely Jacobians), the composition on the right hand side turns into a matrix multiplication. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

A particularly clear formulation of the chain rule can be achieved in the most general setting: let M, N and P be C^k manifolds (or even Banach-manifolds) and let

f : M → N and g : N → P

be differentiable maps. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The derivative of f, denoted by df, is then a map from the tangent bundle of M to the tangent bundle of N, and we may write

$\mbox{d}\left(g \circ f\right) = \mbox{d}g \circ \mbox{d}f.$

In this way, the formation of derivatives and tangent bundles is seen as a functor on the category of C^∞ manifolds with C^∞ maps as morphisms. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

Tensors and the chain rule

See tensor field for an advanced explanation of the fundamental role the chain rule plays in the geometric nature of tensors. In Mathematics, Physics and Engineering, a tensor field is a very general concept of variable geometric quantity History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually

Higher derivatives

Faà di Bruno's formula generalizes the chain rule to higher derivatives. Faà di Bruno's formula is an identity in Mathematics generalizing the Chain rule to higher derivatives named in honor of Francesco Faà di Bruno (1825&ndash1888 The first few derivatives are

$\frac{d (f \circ g) }{dx} = \frac{df}{dg}\frac{dg}{dx}$

$\frac{d^2 (f \circ g) }{d x^2} = \frac{d^2 f}{d g^2}\left(\frac{dg}{dx}\right)^2 + \frac{df}{dg}\frac{d^2 g}{dx^2}$

$\frac{d^3 (f \circ g) }{d x^3} = \frac{d^3 f}{d g^3} \left(\frac{dg}{dx}\right)^3 + 3 \frac{d^2 f}{d g^2} \frac{dg}{dx} \frac{d^2 g}{d x^2} + \frac{df}{dg} \frac{d^3 g}{d x^3}$

$\frac{d^4 (f \circ g) }{d x^4} =\frac{d^4 f}{dg^4} \left(\frac{dg}{dx}\right)^4 + 6 \frac{d^3 f}{d g^3} \left(\frac{dg}{dx}\right)^2 \frac{d^2 g}{d x^2} + \frac{d^2 f}{d g^2} \left\{ 4 \frac{dg}{dx} \frac{d^3 g}{dx^3} + 3\left(\frac{d^2 g}{dx^2}\right)^2\right\} + \frac{df}{dg}\frac{d^4 g}{dx^4}$

References

^ Apostol, Tom (1974). In Calculus, integration by substitution is a tool for finding Antiderivatives and Integrals Using the Fundamental theorem of calculus often requires The triple product rule, known variously as the cyclic Chain rule, cyclic relation, or Euler's chain rule, is a formula which relates In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, Leibniz's rule for Differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an Integral In Calculus, the Leibniz rule, named after Gottfried Leibniz, generalizes the Product rule. Mathematical analysis, 2nd ed. , Addison Wesley, Theorem 5. 5.

External links

Eric W. Weisstein, Chain Rule at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

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