2.4, The Chain Rule

The chain rule allows us to use our derivative rules to compute the derivative of compositions of functions.  It is the most powerful tool in our derivative toolkit and we will spend a week or more ferreting out its power.

The chain rule is motivated by a symbolic simplification,

This should be interpreted as a statement about the derivative of a composition of a function y and a function u. Imagine that y(x) = sin x and u(x) = x^3.  Then the composition of y and u is a new function,  sin(u), or, written in terms of x,

f(x) = sin(x^3). 

Since the derivative of sin x is cos x and the derivative of x^3 is 3x^2 then the derivative of

f(u) = sin(u) 

with respect to u is cos(u) and so the derivative of f(x) with respect to x is

cos(u)(du/dx) = cos(x^3)(3x^2).

This rule allows us to take a complicated composition of functions, such as f(x)=(sin(x^3+e^x))^5, and work out the derivative of the composition based on the individual derivatives of the links in the chain.