NationMaster - Encyclopedia: Chain rule

Topics in calculus
Fundamental theorem Limits of functions Continuity Vector calculus Tensor calculus Mean value theorem Calculus (from Latin, pebble or little stone) is a major area in mathematics where infinitesimal data yields global information. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. ... In mathematics, the limit of a function is a fundamental concept in mathematical analysis. ... 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 multivariate real analysis of vectors in two or more dimensions. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In calculus, the mean value theorem states, roughly, that given a section of a smooth curve, there is a point on that section at which the derivative (slope) of the curve is equal to the average derivative of the section. ...
Differentiation
Product rule Quotient rule Chain rule Implicit differentiation Taylor's theorem Related rates Table of derivatives For a non-technical overview of the subject, see Calculus. ... In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... 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. ... In mathematics, to give a function implicitly is to give an equation that at least in part has the same graph as . ... In calculus, Taylors theorem, named after the mathematician Brook Taylor, who stated it in 1712, gives the approximation of a differentiable function near a point by a polynomial whose coefficients depend only on the derivatives of the function at that point. ... In differential calculus, related rates problems involve ratios of derivatives of two or more related variables that are changing with respect to time. ... The primary operation in differential calculus is finding a derivative. ...
Integration
Lists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution In calculus, the integral of a function is an extension of the concept of a sum. ... See the following pages for lists of integrals: List of integrals of rational functions List of integrals of irrational functions List of integrals of trigonometric functions List of integrals of inverse trigonometric functions List of integrals of hyperbolic functions List of integrals of arc hyperbolic functions List of integrals of... It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ... In mathematics, in particular integral calculus, disk integration (the disk method) is a means of calculating the volume of a solid of revolution. ... Shell integration (the shell method in integral calculus) is a means of calculating the volume of a solid of revolution. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. ...

In calculus, the chain rule is a formula for the derivative of the composite of two functions. Calculus (from Latin, pebble or little stone) is a major area in mathematics where infinitesimal data yields global information. ... In mathematics and in the sciences, a formula (plural: formulae, formulÃ¦ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ... For a non-technical overview of the subject, see Calculus. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

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. In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... Look up Change in Wiktionary, the free dictionary. ... Look up computation in Wiktionary, the free dictionary. ... In mathematics, multiplication is an elementary arithmetic operation. ...

1 Definition
2 Examples
3 Chain rule for several variables
4 Proof of the chain rule
5 The fundamental chain rule
6 Tensors and the chain rule
7 Higher derivatives
8 See also

Definition

The chain rule states that

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

which in short form is written as $(f circ g)' = f'circ gcdot g'$ .

Alternatively, in the Leibniz notation, the chain rule is 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 Î”x and Î”y represent finite...

$frac {df}{dx} = frac {df} {dg} frac {dg}{dx}.$

Note: In text books it is more commonly seen as:

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

In integration, the counterpart to the chain rule is the substitution rule. In calculus, the integral of a function is an extension of the concept of a sum. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ...

Examples

Example I

Suppose, that one is climbing a mountain at a rate of 0.5 kilometres per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 °C per kilometre. If one multiplies 6 °C per kilometre by 0.5 kilometre per hour, one obtains 3 °C per hour. This calculation is a typical chain rule application. Kilometre per hour (American spelling: kilometer per hour) is a unit of both speed (scalar) and velocity (vector). ... Fig. ...

Example II

Consider $f (x) = (x 2 + 1) 3$ . We have $f (x) = h (g (x))$ where $g (x) = x 2 + 1$ and $h (x) = x 3 .$ Thus,

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

In order to differentiate the trigonometric function In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

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

one can write $f (x) = h (g (x))$ with $h (x) = sin x$ and $g (x) = x 2$ . The chain rule then yields

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

since $h'(g (x)) = cos(x 2)$ and $g'(x) = 2 x$ .

Example III

Differentiate $arctan,sin, x$ , etc.

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

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

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

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

${ dz over dt}={partial f over partial x}{dx over dt}+{partial f over partial y}{dy over dt}$

Suppose that each function 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. 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 $vec r$ : In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... For other uses, see Gradient (disambiguation). ...

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

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: In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

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

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. Then by the definition of differentiability,

$g(x+delta)-g(x)= delta g'(x) + epsilon(delta)delta ,$ where $epsilon(delta) to 0 ,$ as $deltato 0.$

Similarly,

$f(g(x)+alpha) - f(g(x)) = alpha f'(g(x)) + eta(alpha)alpha ,$ where $eta(alpha) to 0 ,$ as $alphato 0. ,$

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 $deltato 0,$ $frac{alpha_delta}{delta}to g'(x)$ and $alpha_delta to 0$ , thus $eta(alpha_delta)to 0$ . Therefore

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

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 In mathematics, Banach spaces, 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 mathematics, the FrÃ©chet derivative is a derivative defined on Banach spaces. ...

$mbox{D}_xleft(g circ fright) = mbox{D}_{fleft(xright)}left(gright) circ mbox{D}_xleft(fright).$

Note that the derivatives here are linear maps and not numbers. 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 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 mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... 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 On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

f : M → N and g : N → P

be differentiable maps. 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 In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space...

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

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. Did somebody just say functor? In category theory, 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. ...

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. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

Higher derivatives

Faà di Bruno's formula generalizes the chain rule to higher derivatives. The first few derivatives are // The formula FaÃ di Brunos formula is an identity in mathematics generalizing the chain rule to higher derivatives, named in honor of Francesco FaÃ di Bruno (1825â€“1888), who was (in chronological order) a military officer, a mathematician, and a priest, and was beatified by the Pope a century...

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

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

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

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

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