5.2.3 Higher 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

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   

• • 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