## Differentiation

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 axn = naxn − 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 x3 6x4 5x 23 Answer 3x² 24x3 5 0

### Differentiation of Polynomial functions

A polynomial function is differentiated term by term.

Example

If f(x) = 2x3 + 3x2 − 4x + 6

then f '(x) = 6x2 + 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  