Chain rule

In calculus, 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

In Leibniz's notation, the Chain Rule can be expressed as:

or

Intuitively, 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.

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

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

Table of contents

1 Example I 2 Example II 3 Proof of Chain rule 4 The Fundamental Chain Rule 5 Tensors and the chain rule as cocycle

Example I

Consider:

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

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.

Proof of Chain rule

Suppose functions f(x) and g(x) are continuous and differentiable. We let function

Using the definition of the derivative of a function:

By the Mean Value Theorem (MVT), there exist some c satisfying

such that

By subsituting (2) into (1), we get

Therefore

as required.

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

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

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 as cocycle

As an advanced explanation of the tensor concept, one can interpret the chain rule as applied to coordinate changes also as the requirement for self-consistent concepts of tensor giving rise to tensor fields. Abstractly, we can identify the chain rule as a cocycle. It gives the consistency required to define the tangent bundle in an intrinsic way. The other vector bundles of tensors have comparable cocycles, that come from applying functorial properties of tensor constructions to the chain rule itself; which is why they also are intrinsic (read, 'natural') concepts. What can be read as the 'classical' approach to tensors tries to read this backwards - and is therefore a heuristic approach rather than a foundational one.