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:
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The derivative of axn = naxn − 1
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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."
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Example 1
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Example 2
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Example 3
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Example 4
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| Differentiate |
x3
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6x4
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5x
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23
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| Answer |
3x²
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24x3
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5
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0
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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
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Example 1
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Example 2
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