Differentiation from first principles is a long process and most differentiation can be done by using rules and learning the basic derivatives.
Polynomial expressions and power functions can be differentiated term by term according to the rule:
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If f (x) = axn
then f ' (x) = naxn − 1
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In words "multiply the coefficient by the exponent and then lower the exponent by 1."
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There are two other ways to indicate the derived function.
and y ' both mean the same as f '(x).
Examples of differentiation of polynomials
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Function
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Derivative
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| f(x) = x3 |
f '(x) = 3x2 |
| g(x) = 4x2 + 3x + 2 |
g '(x) = 8x + 3 |
| y = 6x4 − 5 |
y ' = 24x3 |
| f(x) = (x − 3)(x + 2) = x2 − x − 6 |
f '(x) = 2x − 1 |
More difficult examples of differentation including power functions
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Function
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Derivative
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f(x) = √x
= x 0.5
= 
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y = 
= 5x -2
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 = 8x − 2 |
IMPORTANT In most of the above examples the functions have to be written in index form or expanded before differentiation.
