Chain rule

Topics in calculus

Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials | Mean value theorem | Vector calculus | Tensor calculus

Differentiation

Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem | Related rates

Integration

Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Improper integrals | Lists of integrals

In calculus, the chain rule is a formula for the derivative of the composition of two functions.

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 the function f is differentiable at g(x) and the function g is differentiable at x, and the function F is defined as f composed with g, that is

<math>

F = f \circ g = f(g(x))

<math>

then <math>F'<math> is given by

<math>

F' = \frac {dF} {dx} = f'(g(x)) \times g'(x).

<math>

Alternatively, in Leibniz notation, the chain rule can be expressed as:

<math>

\frac {dy}{dx} = \frac {dy} {du} \times \frac {du}{dx} <math> or

<math>

\frac {d(f \circ g)}{dx} = \frac {d(f \circ g)} {dg} \times \frac {dg}{dx}. <math>

Contents

1 The general power rule 1.1 Example I 1.2 Example II 2 Proof of the chain rule 3 The fundamental chain rule 4 Tensors and the chain rule

The general power rule

The general power rule (GPR) is derivable, via the Chain Rule.

Example I

Consider:

<math>f\left(x\right) = \left(x^2 + 1\right)^3<math>

f(x) is comparable to h[g(x)] where g(x) is (x² + 1) and h(x) is x³; thus,

<math>f'\left(x\right) = 3\left(x^2 + 1\right)^2\left(2x\right) = 6x\left(x^2 + 1\right)^2.<math>

Example II

In order to differentiate the trigonometric function:

f(x) = sin(x²)

one can write f(x) = h(g(x)) with h(x) = sin(x) 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.

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,

<math> g(x+\delta)-g(x)= \delta g'(x) + \epsilon(\delta)<math> where <math> \epsilon(\delta)/\delta\to 0<math> as <math>\delta\to 0.<math>

Similarly,

<math> f(g(x)+\alpha) - f(g(x)) = \alpha f'(g(x)) + \eta(\alpha)<math> where <math>\eta(\alpha)/\alpha \to 0<math> as <math>\alpha\to 0.<math>

Now

<math> f(g(x+\delta))-f(g(x)) = f(g(x) + \delta g'(x)+\epsilon(\delta)) - f(g(x))<math>

<math> = \alpha_\delta f'(g(x)) + \eta(\alpha_\delta)<math>

where <math>\alpha_\delta = \delta g'(x) + \epsilon(\delta)<math>. Observe that as <math>\delta\to 0,<math> <math>\alpha_\delta/\delta\to g'(x)<math> and <math>\eta(\alpha_\delta)/\delta\to 0<math>. Hence

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

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

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

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.

A particularly nice 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. 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

<math>\mbox{d}\left(g \circ f\right) = \mbox{d}g \circ \mbox{d}f.<math>

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.

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.

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