## 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 d^{2}y/dx^{2}, 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.

If |
d^{2}y |
is positive, then it is a minimum point |

dx^{2} |

If |
d^{2}y |
is negative, then it is a maximum point |

dx^{2} |

If |
d^{2}y |
= zero, then it could be a maximum, minimum or point of inflexion |

dx^{2} |

If d^{2}y/dx^{2} = 0, you must test the values of dy/dx either side of the stationary point, as before in the stationary points section.

**Example**

Find the stationary points on the curve y = x^{3} - 27x and determine the nature of the points:

At stationary points, dy/dx = 0

dy/dx = 3x^{2} - 27

If this is equal to zero, 3x^{2} - 27 = 0

Hence x^{2} - 9 = 0 (dividing by 3)

So (x + 3)(x - 3) = 0

So x = 3 or -3

d^{2}y/dx^{2} = 6x

When x = 3, d^{2}y/dx^{2} = 18, which is positive.

When x = -3, d^{2}y/dx^{2} = -18, which is negative.

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