**The Chain Rule **The chain rule is very important in differential calculus and states that:

dy = dy × dt

dx dt dx

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. This is because:

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-1) multiplied by the contents of the bracket raised to the power of (n-1).

**The Product Rule **This is another very useful formula:

d (uv) = vdu + udv

dx dx dx

Example:

Differentiate x(x² + 1)

let u = x and v = x² + 1

d (uv) = (x² + 1) + x(2x) = x² + 1 + 2x² = 3x² + 1 .

dx

Again, with practise you shouldn't have to write out u = ... and v = ... every time.

**The Quotient Rule**

d (u/v) = v(du/dx) - u(dv/dx)

dx v²

Example:

If y = x³ , find dy/dx

x + 4

Let u = x³ and v = (x + 4). Using the quotient rule, dy/dx =

(x + 4)(3x²) - x³(1) = 2x³ + 12x²

(x + 4)² (x + 4)²