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 $y=x^4+6x^2$ .

Solution

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

$\text{first derivative}=4x^3+12x$

To obtain the second derivative we differentiate the first derivative.

$\text{second derivative}=12x^2+12$

The third derivative is found by differentiating the second derivative.

$\text{third derivative}=24x+0=24x$