Encyclopedia > Chain rule

  Article Content

Chain rule

In calculus, the Chain Rule states; 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; times, the rate of change of u, with respect to x. Suppose one is climbing a mountain, at a rate of 0.5 kilometers per hour. The temperature is lower at higher elevations; suppose the rate, by which it decreases, is 6° per kilometer. How fast does the temperature drop? Well, if one multiplies 6° per kilometer, by 0.5 kilometers per hour; one obtains 3° per hour. Such calculations are the "heart" of the Chain Rule.

Leibniz would express the Chain Rule as:

dy/dx = (dy/du) · (du/dx)

The Chain Rule is a formula for the derivative of the composition of two functions. Suppose the real-valued function g(x) is defined on some open subset, of the real numbers, containing the number x; and h[g(x)] is defined on some open subset of the reals containing g(x). If g is differentiable at x and h is differentiable at g(x), then the composition h o g is differentiable at x and the derivative can be computed as

f '(x) = (h o g)'(x) = h '[g(x)] · g '(x)

Table of contents

Example I Consider f(x) = (x2 + 1)3. f(x) is comparable to h[g(x)] where g(x) is (x2 + 1) and h(x) is x3; thus, f '(x) = 3(x2 + 1)2(2x) = 6x(x2 + 1)2.

Example II In order to differentiate the trigonometric function:

f(x) = sin(x2)
one can write f(x) = h(g(x)) with h[f(x)] = sin(x2) and g(x) = x2 and the chain rule then yields
f '(x) = cos(x2) 2x
since h '[g(x)] = cos(x2) and g '(x) = 2x.

The General Power Rule The General Power Rule (GPR) is derivable, via the Chain Rule.

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 of the composition g o f at the point x is given by

Dx(g o f) = Df(x)(g) o Dx(f)
Note that the derivatives here are linear maps and not numbers. If the linear maps are represented as matrices, 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 manifolds (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 bundle of M to the tangent bundle of N, and we may write

d(g o f) = dg o df
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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Political parties of the world

... Partido Verde Ecologista (PVEM) (Green Party) Historic Parties: Conservativos Netherlands Main article: List of political parties in the Netherlands ...