The Second Derivative

The second derivative is what you get when you differentiate the derivative. Remember that the derivative of y with respect to x is written dy/dx. The second derivative is written d2y/dx2, pronounced "dee two y by d x squared".

second derivative

Stationary Points

The second derivative can be used as an easier way of determining the nature of stationary points (whether they are maximum points, minimum points or points of inflection).

A stationary point on a curve occurs when dy/dx = 0. Once you have established where there is a stationary point, the type of stationary point (maximum, minimum or point of inflexion) can be determined using the second derivative.

If d2y is positive, then it is a minimum point
If d2y is negative, then it is a maximum point
If d2y = zero, then it could be a maximum, minimum or point of inflexion

If d2y/dx2 = 0, you must test the values of dy/dx either side of the stationary point, as before in the stationary points section.


Find the stationary points on the curve y = x3 - 27x and determine the nature of the points:

At stationary points, dy/dx = 0
dy/dx = 3x2 - 27

If this is equal to zero, 3x2 - 27 = 0
Hence x2 - 9 = 0 (dividing by 3)
So (x + 3)(x - 3) = 0
So x = 3 or -3

d2y/dx2 = 6x
When x = 3, d2y/dx2 = 18, which is positive.
When x = -3, d2y/dx2 = -18, which is negative.

Hence there is a minimum point at x = 3 and a maximum point at x = -3.