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".

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.

Example

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.

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