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
- a second order derivative is the rate of change of the rate of change of a function
- 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
Exam Tip
- Even if you think you can find the second derivative in your head and write it down, make sure you write down the first derivative as well
- If you make a mistake, you will most likely get marks for finding the first derivative
Worked example
Work out when
Find the first derivative of the function first by considering each term in turn.
Find the second derivative, by differentiating each term in the first derivative.
To find the first derivative of the function, begin by separating the terms in the fraction.
Rewrite each term using index notation so that they are in a form that can be differentiated.
Find the first derivative of the function first by differentiating each term in turn.
Find the second derivative, by differentiating each term in the first derivative.
You can turn the second derivative back into the same format as the original function by rewriting as a fraction.