Chain Rule

The chain rule is an approach for differentiating a composition of functions in terms of those functions.

Contents

Notation

The common notation for the chain rule is, given a function h(x) = f(g(x)), the derivative h'(x) = f'(g(x)) g'(x).

An equivalent notations is that h = f ∘ g and h' = (f ∘ g)' = (f' ∘ g)g' .

A parallel notation is:

It can be simplified using the chain rule into:

Start with the final formulation and use u- and v-substitution:

Note in particular that the equation for v can be rewritten in terms of P:

Now the final formulation can be rewritten as:

Several of these terms are trivial to derive.