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, that is we have $f \circ g = f(g(x))$ . Then

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

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

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

where $\frac {df} {dg}$ indicates f depends on g as if it were a variable.

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

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)

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 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 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