Second Derivatives (AQA GCSE Further Maths): Revision Note

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 $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$

• Note the position of the powers of 2

  • differentiating twice (so${\mathbf{d}}^{\mathbf{2}}$) with respect to $x$ twice (so${\mathit{x}}^{\mathbf{2}}$)

• 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

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

  • A negative second derivative means the gradient is decreasing

    • 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

    • $y=5x^3+10x^2$

    • $\frac{dy}{dx}=15x^2+20x$

    • $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}=30x+20$