IBMaths: AI HLDPRevision Notes5. Calculus5.2 Further Differentiation5.2.4 Second Order Derivatives

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
- $y = f (x)$
- $\frac{d y}{d x} = f' (x)$ (First order derivative)
- $\frac{d^{2} y}{d x^{2}} = f'' (x)$ (Second order derivative)
Note the position of the superscript 2’s
- differentiating twice (so $d^{2}$ ) with respect to $x$ twice (so $x^{2}$ )
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
- a 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

Exam Tip

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

Worked example

Given that $f (x) = 4 - \sqrt{x} + \frac{3}{\sqrt{x}}$

Find

f' (x)

and

f'' (x)

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

Evaluate

f'' (3)

.
Give your answer in the form

a \sqrt{b}

, where

b

is an integer and

a

is a rational number.

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

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Author: Paul

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.