## 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.

**Example**

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).