ASMathsEdexcelPureRevision NotesDifferentiationDifferentiationDifferentiating Powers of x

Differentiating Powers of x (Edexcel AS Maths): Revision Note

Exam code: 8MA0

Roger B

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Roger B

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Edexcel Maths: PureCambridge (CIE) Maths: Pure 1AQA Maths: PureOCR Maths A: Pure

Differentiating powers of x

How do I differentiate expressions with powers of x?

  • Powers of x are differentiated according to the following formula

    • fraction numerator d over denominator d x end fraction open parentheses x to the power of n close parentheses equals n x to the power of n minus 1 end exponent

      • Where n is any real constant

    • This can be written as:

      • If straight f open parentheses x close parentheses equals x to the power of n then straight f apostrophe open parentheses x close parentheses equals n x to the power of n minus 1 end exponent

      • If y equals x to the power of n then fraction numerator d y over denominator d x end fraction equals n x to the power of n minus 1 end exponent

Examiner Tips and Tricks

straight f apostrophe open parentheses x close parentheses is usually used when a function, straight f open parentheses x close parentheses, has been defined. fraction numerator d y over denominator d x end fraction is used when a graph of a function is given in the form y equals... They both represent the derivative of an expression.

  • If the term involves a scalar multiple, then you can differentiate the power of x and then multiply by the scalar

    • fraction numerator d over denominator d x end fraction open parentheses a x to the power of n close parentheses equals a n x to the power of n minus 1 end exponent

      • Where n and a are any real constants

  • Differentiating terms with positive integer powers of x is straightforward

Pwrs of x Illustr 2, A Level & AS Maths: Pure revision notes

  • But negative and fractional powers of x can also be differentiated this way

Pwrs of x Illustr 3, A Level & AS Maths: Pure revision notes

 

  • Don't forget these two special cases:

    • The derivative of a linear function is a constant

      • fraction numerator d over denominator d x end fraction open parentheses a x close parentheses equals a

      • Where a is any real constant

    • The derivative of a constant function is zero

      • fraction numerator d over denominator d x end fraction open parentheses a close parentheses equals 0

      • Where a is any real constant

  • These allow you to differentiate constants and linear terms in x

Pwrs of x Illustr 4b, A Level & AS Maths: Pure revision notes

  • If the expression involves a sum or difference of terms just differentiate one term at a time

Pwrs of x Illustr 5, A Level & AS Maths: Pure revision notes

Examiner Tips and Tricks

  • Take extra care when differentiating negative and fractional powers of x as mishandling negative signs and fractions is a common way to lose marks in these sorts of questions.

Worked Example

Pwrs of x Example, A Level & AS Maths: Pure revision notes

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Roger B

Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.