When you are given a function

and you find

this process is called differentiation.

We have, until now, used the definition

to calculate $f^{\prime }(x)$ and, whilst this is completely correct and rigorous, it is slow and difficult.

In this chapter, we are going to differentiate polynomials (and beyond) using the formula

This formula is proved by the first principles definition - and now we have the formula we never have to use first principles again! (Except in examinations.)

Let's see an example of the formula at work.

1. Given that

   $$y=x^2(\sqrt{x}+x^3)$$

   find

   $$\frac{dy}{dx}$$

2. Given that

   $$v=\frac{\sqrt[3]{u}+u^{\frac{5}{2}}}{u^9}$$

   find

   $$\frac{dv}{du}$$