### 1 The derivative of a derivative

You have already learnt how to calculate the derivative of a function using a table of derivatives and applying some basic rules. By differentiating the function, $y\left(x\right)$ , we obtain the derivative, $\frac{dy}{dx}$ . By repeating the process we can obtain higher derivatives.

##### Example 7

Calculate the first, second and third derivatives of $y={x}^{4}+6{x}^{2}$ .

##### Solution

The first derivative is $\frac{dy}{dx}$ :

$\phantom{\rule{2em}{0ex}}\text{firstderivative}=4{x}^{3}+12x\phantom{\rule{1em}{0ex}}$

To obtain the second derivative we differentiate the first derivative.

$\phantom{\rule{2em}{0ex}}\text{secondderivative}=12{x}^{2}+12$

The third derivative is found by differentiating the second derivative.

$\phantom{\rule{2em}{0ex}}\text{thirdderivative}=24x+0=24x$