Chain Rule

From Exampleproblems

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 kilometres per hour. The temperature is lower at higher elevations; suppose the rate by which it decreases is 6 °F per kilometre. If one multiplies 6 °F per kilometre by 0.5 kilometre per hour, one obtains 3 °F 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. The chain rule may be stated in any of several equivalent forms:

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

or in the Leibniz notation

$\frac {df}{dx} = \frac {df} {dg} \cdot \frac {dg}{dx},$

$\frac {df} {dx} = \frac {d} {dx} f(g(x)) = f'(g(x)) g'(x).$

In integration, the counterpart to the chain rule is the substitution rule.

1 The general power rule
- 1.1 Example I
- 1.2 Example II
2 Chain rule for several variables
3 Proof of the chain rule
4 The fundamental chain rule
5 Tensors and the chain rule
6 Higher derivatives

The general power rule

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

Example I

Consider $f (x) = (x 2 + 1) 3$ . $f (x)$ is comparable to $h (g (x))$ where $g (x) = x 2 + 1$ and $h (x) = x 3$ ; thus,

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

Example II

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 $g (x) = x 2$ . The chain rule then yields

f'(x) = 2 x cos(x 2)

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

Chain rule for several variables

The chain rule works for functions of several variables as well. For example, if we have a function $f (u (x, y), v (x, y))$ where

u (x, y) = 3 x + y 2

and

v (x, y) = sin(x y)

, and if

f = u + v

then

${\partial f \over \partial x}={\partial f \over \partial u}{\partial u \over \partial x}+{\partial f \over \partial v}{\partial v \over \partial x}=3 + \cos(xy)y.$

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,

$g(x+\delta)-g(x)= \delta g'(x) + \epsilon(\delta) \,$ where $\frac{\epsilon(\delta)}{\delta} \to 0 \,$ as $\delta\to 0.$

Similarly,

$f(g(x)+\alpha) - f(g(x)) = \alpha f'(g(x)) + \eta(\alpha) \,$ where $\frac{\eta(\alpha)}{\alpha} \to 0 \,$ as $\alpha\to 0. \,$

Now

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

where $\alpha_\delta = \delta g'(x) + \epsilon(\delta) \,$ . Observe that as $\delta\to 0,$ $\frac{\alpha_\delta}{\delta}\to g'(x)$ and $\frac{\eta(\alpha_\delta)}{\delta}\to 0$ . Hence

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

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. If the linear maps are represented as matrices (namely Jacobians), the composition on the right hand side turns into a matrix multiplication.

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. 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.

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.

Higher derivatives

Faà di Bruno's formula generalizes the chain rule to higher derivatives. The first few derivatives are

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

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

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

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