Higher Order Derivatives (DP IB Analysis & Approaches (AA)): Revision Note

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

  • 

  •      (First order derivative) 

  •      (Second order derivative)

• Note the position of the superscript 2’s

  • differentiating twice (so) with respect to  twice (so)

• 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

  • help 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

Higher Order Derivatives

What is meant by higher order derivatives of a function?

• Many functions can be differentiated numerous times

  • The third, fourth, fifth, etc derivatives of a function are generally called higher order derivatives

• It may not be possible, or practical to (algebraically) differentiate complicated functions more than once or twice

• Polynomials will, eventually, have higher order derivatives of zero

  • Since powers of x reduce by 1 each time

What is the notation for higher order derivatives?

• The notation for higher order derivatives follows the logic from the first and second derivatives

   or   

except the ‘dash’ (prime) notation is replaced with numbers as this would become cumbersome after the first few

• e.g. the fifth derivative would be

    or  

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

• By differentiating as many times as required!

• 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