Second Order Derivatives (DP IB Maths: AI HL)

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Second Order Derivatives

What is the second order derivative of a function?

  • If you differentiate the derivative of a function (i.e. differentiate the function a second time) you get the second order derivative of the function
  • There are two forms of notation for the second order derivative
    • space y equals f left parenthesis x right parenthesis
    • space fraction numerator straight d y over denominator straight d x end fraction equals f apostrophe left parenthesis x right parenthesis     (First order derivative) 
    • space fraction numerator straight d squared y over denominator straight d x squared end fraction equals f apostrophe apostrophe left parenthesis x right parenthesis     (Second order derivative)
  • Note the position of the superscript 2’s
    • differentiating twice (sobold italic space bold d to the power of bold 2) with respect to x twice (sobold space bold italic x to the power of bold 2)
  • The second order derivative can be referred to simply as the second derivative
    • Similarly, the first order derivative can be just the first derivative
  • A first order derivative is the rate of change of a function
    • second order derivative is the rate of change of the rate of change of a function
      • i.e. the rate of change of the function’s gradient
  • Second order derivatives can be used to

        • test for local minimum and maximum points
        • help determine the nature of stationary points
        • determine the concavity of a function
        • graph derivatives

How do I find a second order derivative of a function?

  • By differentiating twice!
  • This may involve
    • rewriting fractions, roots, etc as negative and/or fractional powers
    • differentiating trigonometric functions, exponentials and logarithms
    • using chain rule
    • using product or quotient rule

Examiner Tip

  • Negative and/or fractional powers can cause problems when finding second derivatives so work carefully through each term

Worked example

Given that space f left parenthesis x right parenthesis equals 4 minus square root of x plus fraction numerator 3 over denominator square root of x end fraction

a)
Findspace f apostrophe left parenthesis x right parenthesis andspace f apostrophe apostrophe left parenthesis x right parenthesis.

5-2-3-ib-sl-aa-only-second-order-we-soltn-a

b)
Evaluatespace f apostrophe apostrophe left parenthesis 3 right parenthesis.
Give your answer in the form a square root of b, where space b is an integer and space a is a rational number.

5-2-3-ib-sl-aa-only-second-order-we-soltn-b

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.