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ASMaths: Pure 1CIERevision Notes5. Differentiation5.2 Applications of Differentiation5.2.3 Second Order Derivatives

Second Order Derivatives (CIE AS Maths: Pure 1)

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Second Order Derivatives

What is the second order derivative of a function?

  • If you differentiate the derivative of a function (ie differentiate the function a second time) you get the second order derivative of the function
  • For a function y = f(x), there are two forms of notation for the second derivative (or second order derivative)

straight f to the power of apostrophe apostrophe end exponent left parenthesis x right parenthesis or fraction numerator d squared y over denominator d x squared end fraction

    • Note the positions of the power of 2's in the second version 

2nd Order Deriv Illustr 2, AS & A Level Maths revision notes

  • The second order derivative can also be referred to simply as the second derivative
    • Similarly, the 'regular' derivative can also be referred to as either the first order derivative or the first derivative
  • The second order derivative gives the rate of change of the gradient function (ie of the first derivative) – this will be important for identifying the nature of stationary points

Examiner Tip

  • When finding second derivatives be especially careful with functions that have negative or fractional powers of x (see Worked Example below).
  • Mistakes made with fractions or negative signs can build up as you calculate the derivative more than once.

Worked example

2nd Order Deriv Example, A Level & AS Maths: Pure revision notes

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    Roger

    Author: Roger

    Expertise: Maths

    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.