The process of finding the gradient or derived function is called differentiation. There is a way of doing this called differentiating from first principles and this is studied in more detail in Year 12.
A quicker way to differentiate polynomial funtions and terms is to use a formula or rule.
For a function f(x) the derived function is given by f '(x).
An alternative notation is that for a function y the derived function is y ' or
The rule is:
The derivative of ax^{n} = nax^{n − 1}

In words this says that to find the derivative of a term:
"multiply the coefficient by the exponent and then lower the exponent by 1."
Example 1

Example 2

Example 3

Example 4


Differentiate 
x^{3}

6x^{4}

5x

23

Answer 
3x²

24x^{3}

5

0

Differentiation of Polynomial functions
A polynomial function is differentiated term by term.
Example
If f(x) = 2x^{3} + 3x^{2} − 4x + 6
then f '(x) = 6x^{2} + 6x − 4
A function containing brackets must be expanded before differentiating.
Example
If y = (x + 2)(x − 3)
Expanding y = x² − x − 6
= 2x − 1
Differentiation of Fractional Indices (Roots)
Roots such as √x can be written as . This can now be differentiated as normal.
If f(x) = √x =
Differentiation of Negative Indices
Terms such as have to be written as x ^{2}. This can now be differentiated as normal.
If y = = x ^{2}
Differentiation of Rational Expressions
Expressions involving fractions may need simplifiying or cancelling first
Example 1

Example 2
