Second Order Derivatives (Cambridge International AS Mathematics)

Second order derivatives

What is the second order derivative of a function?

• If you differentiate the derivative of a function (ie differentiate the function a second time) you get the second order derivative of the function
• For a function y = f(x), there are two forms of notation for the second derivative (or second order derivative)

$f^{''}(x)$ or $\frac{d^{2}y}{dx^{2}}$

• Note the positions of the power of 2's in the second version

• The second order derivative can also be referred to simply as the second derivative

  • Similarly, the 'regular' derivative can also be referred to as either the first order derivative or the first derivative

• The second order derivative gives the rate of change of the gradient function (ie of the first derivative) – this will be important for identifying the nature of stationary points

Exam Hint

• When finding second derivatives be especially careful with functions that have negative or fractional powers of x (see Worked Example below).
• Mistakes made with fractions or negative signs can build up as you calculate the derivative more than once.

Worked Example