The Chain Rule

The chain rule (function of a function) is very important in differential calculus and states that:

dy = dy × dt
dx   dt   dx

(You can remember this by thinking of dy/dx as a fraction in this case (which it isn’t of course!)).

This rule allows us to differentiate a vast range of functions.


If y = (1 + x²)³ , find dy/dx .
let t = 1 + x²
therefore, y = t³
dy/dt = 3t²
dt/dx = 2x
by the Chain Rule, dy/dx = dy/dt × dt/dx
so dy/dx = 3t² × 2x = 3(1 + x²)² × 2x
= 6x(1 + x²)²

In examples such as the above one, with practise it should be possible for you to be able to simply write down the answer without having to let t = 1 + x² etc.

In other words, the differential of something in a bracket raised to the power of n is the differential of the bracket, multiplied by n times the contents of the bracket raised to the power of (n-1).