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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 product of the rate of change of y with respect to u multiplied by the rate of change of u with respect to x. Suppose, for example, that one is climbing a mountain at a rate of 0.5 kilometre per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6° per kilometre. How fast does the temperature drop? Well, if one multiplies 6° per kilometre by 0.5 kilometre per hour, one obtains 3° per hour. This calculation is a typical chain rule application.
In algebraic terms, the chain rule (of one variable) states that if functions f and g are both differentiable and function F is defined as f composed with g, that is
then is given by
Alternatively, in Leibniz notation, the chain rule can be expressed as:
or
The general power rule (GPR) is derivable, via the Chain Rule.
Consider:
f(x) is comparable to h[g(x)] where g(x) is (x2 + 1) and h(x) is x3; thus,
In order to differentiate the trigonometric function:
one can write f(x) = h(g(x)) with h(x) = sin(x) and g(x) = x2 and the chain rule then yields
since h '[g(x)] = cos(x2) and g '(x) = 2x.
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,
Similarly,
Now
where . Observe that as and . Hence
as .
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 spaceFunctional analysis In mathematics, Banach spaces named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions. Definition Banach ss (which includes Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers) 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 of the composition g o f at the point x is given by
Note that the derivatives here are linear mapsIn mathematics, a linear transformation (also called linear operator or linear map is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it "prese and not numbers. If the linear maps are represented as matricesAbstract algebra Algebra Linear algebra In mathematics, a matrix (plural matrices is a rectangular table of numbers or, more generally, of elements of a fixed ring. In this article, if unspecified, the entries of a matrix are always real or complex number (namely JacobianIn vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant . Also, in algebraic geometry the Jacobian of a curve means the Jacobian variety: a group structure, which can be imposed on the curvs), the composition on the right hand side turns into a matrix multiplication.
A particularly nice formulation of the chain rule can be achieved in the most general setting: let M, N and P be Ck manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefors (or even Banach-manifolds) and let f : M -> N and g : N -> P be differentiable maps. The derivative of f, denoted by df, is then a map from the tangent bundleIn mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. The tangent bundle of manifold M is usually denot of M to the tangent bundle of N, and we may write
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.
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