Second Derivatives (AQA GCSE Further Maths)

Second 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

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

• We can write the second derivative as 

• Note the position of the powers of 2

  • differentiating twice (so) with respect to  twice (so)

• A first derivative is the rate of change of a function (the gradient)

  • 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

  • A positive second derivative means the gradient is increasing (graph is becoming steeper)

    • For instance in a u-shape, where the gradient is changing from negative to positive

  • A negative second derivative means the gradient is decreasing (graph is becoming less steep)

    • For instance in an n-shape, where the gradient is changing from positive to negative

• Second order derivatives can be used to test whether a point is a minimum or maximum

• To find a second derivative, you simply differentiate twice!

  • It is important to write down your working with the correct notation, so you know what each expression means

  • For example

    • 

    • 

    •