Definition of successive derivatives

We have seen that the derivative of a function of $x$ is in general also a function of $x$. This new function may also be differentiable, in which case the derivative of the first derivative is called the second derivative of the original function. Similarly, the derivative of the second derivative is called the third derivative; and so on to the $n$-th derivative. Thus, if

$$\begin{array}{rl} \ y &=\ 3x^4,\\ \frac{dy}{dx} &=\ 12x^3,\... ...} \left ( \frac{dy}{dx} \right ) \right ] &=\ 72x, \end{array}$$

and so on.