Higher-Order Derivatives (College Board AP® Calculus BC) : Study Guide

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

Second derivatives

What is a second derivative?

  • The second derivative of a function is the derivative, differentiated

    • I.e. it is the derivative of the derivative

    • It may also be referred to as the second differential

  • The second derivative may be written as:

    • f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses

    • fraction numerator d squared y over denominator d x squared end fraction

    • y to the power of apostrophe apostrophe end exponent

  • The second derivative is the rate of change of the rate of change

    • E.g. If a function is increasing (first derivative),

      • the second derivative describes how rapidly its rate of increase is increasing (or decreasing)

  • Second derivatives are useful when investigating the shapes of the graphs of functions

    • See the study guide on 'Concavity of Functions' for more about this

  • To find a second derivative, simply differentiate the function, and then differentiate it again

    • The function and its first derivative must be differentiable in order to do this

Worked Example

A function f is defined by f open parentheses x close parentheses equals 3 x cubed minus 2 x squared minus 3 x plus 2.

Find the second derivative of f open parentheses x close parentheses.

Answer:

Differentiate f open parentheses x close parentheses to find the first derivative

f to the power of apostrophe open parentheses x close parentheses equals 9 x squared minus 4 x minus 3

Differentiate f to the power of apostrophe open parentheses x close parentheses to find the second derivative

f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses equals 18 x minus 4

Higher-order derivatives

What are higher-order derivatives?

  • Extending the idea of second derivatives, the function can continue to be differentiated multiple times

    • This applies as long as the derivatives continue to be differentiable

  • The first, second, third and fourth derivatives may be written as

    • f to the power of apostrophe open parentheses x close parentheses comma space f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses comma space f to the power of open parentheses 3 close parentheses end exponent open parentheses x close parentheses comma space f to the power of open parentheses 4 close parentheses end exponent open parentheses x close parentheses

    • fraction numerator d y over denominator d x end fraction comma space fraction numerator d squared y over denominator d x squared end fraction comma space fraction numerator d cubed y over denominator d x cubed end fraction comma space fraction numerator d to the power of 4 y over denominator d x to the power of 4 end fraction

    • y to the power of apostrophe comma space y to the power of apostrophe apostrophe end exponent comma space y to the power of open parentheses 3 close parentheses end exponent comma space y to the power of open parentheses 4 close parentheses end exponent

      • You may occasionally see f to the power of apostrophe apostrophe apostrophe end exponent open parentheses x close parentheses or y to the power of apostrophe apostrophe apostrophe end exponent for the third derivative

      • But beyond the third derivative, the 'tick mark' notation is almost never used

  • The third derivative is the:

    • rate of change, of the rate of change, of the rate of change of a function

    • This can be extended for higher derivatives if needed

  • A common use of higher order derivatives is when describing motion

    • If a function describes the displacement of an object,

      • the first derivative describes its velocity

      • the second derivative describes its acceleration

      • the third derivative describes the rate of change of acceleration (sometimes referred to as 'jerk')

  • Another common use is when applying L'Hospital's Rule

    • limit as x rightwards arrow a of fraction numerator f left parenthesis x right parenthesis over denominator g left parenthesis x right parenthesis end fraction equals limit as x rightwards arrow a of fraction numerator f to the power of apostrophe left parenthesis x right parenthesis over denominator g to the power of apostrophe left parenthesis x right parenthesis end fraction equals limit as x rightwards arrow a of fraction numerator f to the power of apostrophe apostrophe end exponent left parenthesis x right parenthesis over denominator g to the power of apostrophe apostrophe end exponent left parenthesis x right parenthesis end fraction equals space...

    • As long as the limits continue giving indeterminate forms you can continue applying L’Hospital’s rule with higher order derivatives

    • This can make limits far simpler to evaluate

Worked Example

(a) Find the third derivative of f open parentheses x close parentheses equals 4 x cubed minus 3 x squared plus 9 x minus 8.

Answer:

Find the first derivative

f to the power of apostrophe open parentheses x close parentheses equals 12 x squared minus 6 x plus 9

Find the second derivative

f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses equals 24 x minus 6

Find the third derivative

f to the power of open parentheses 3 close parentheses end exponent open parentheses x close parentheses equals 24

(b) Given that y equals sin space x, find fraction numerator d to the power of 4 y over denominator d x to the power of 4 end fraction.

Answer:

This is asking for the fourth derivative of sin space x

You might be able to write this answer down straight away if you are familiar with derivatives of trig functions!

Find the first derivative

fraction numerator d y over denominator d x end fraction equals cos space x

Find the second derivative

fraction numerator d squared y over denominator d x squared end fraction equals negative sin space x

Find the third derivative

fraction numerator d cubed y over denominator d x cubed end fraction equals negative cos space x

Find the fourth derivative

fraction numerator d to the power of 4 y over denominator d x to the power of 4 end fraction equals sin space x

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Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

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