What is the chain rule?

Up until now you may have been given functions like f(x)= 7x^3 +cos(x) and been told just to differentiate. What your teacher really meant was to differentiate with respect to x. The chain rule allows us to differentiate more complicated functions built out of more simple functions which you can differentiate. It builds on and makes use of what you already know. Specifically, it allows you to differentiate functions of functions (remember composite functions from GCSE!) The chain rule can be written several ways. One such way is: df/dx = (du/dx)df/du. Let’s take an example f(x) = sin(x^2). We can see that f(x) is a composite of two functions which we are familiar with and know how to differentiate. I.e. f(x) = sin(u) if u = x^2. We then have df/du = cos(u) and du/dx = 2x. Using our chain rule df/dx = 2xcos(u). We could leave it there but it would look nicer if we had it all in x but we know u = x^2. Hence our final answer is df/dx = 2x*cos(x^2). There we have it, we have differentiated a function which might have looked more complicated at first by breaking it up into two functions which we already know how to differentiate. This is what the chain rule is all about: making our life easier.