We can differentiate a function many times. The $nth$ derivative is represented by the notation: $$ \begin{equation}\begin{aligned} \frac{d^ny}{dx^n} \text{ OR }f^{(n)}(x)\\ \end{aligned}\end{equation} $$
For example, the second derivative is written as $$ \begin{equation}\begin{aligned} \frac{d^2y}{dx^2}\\ \end{aligned}\end{equation} $$
Example Find the third derivative of the function $y=\cos{3x}$
Solution The first derivative is $$ \begin{equation}\begin{aligned} \frac{dy}{dx}&=-\sin{3x}\times 3\\ &=-3\sin{3x}\\ \end{aligned}\end{equation} $$
The second derivative is $$ \begin{equation}\begin{aligned} \frac{d^2y}{dx^2}&=-3\cos{3x}\times 3\\ &=-9\cos{3x}\\ \end{aligned}\end{equation} $$
The third order derivative is $$ \begin{equation}\begin{aligned} \frac{d^3y}{dx^3}&=9\sin{3x}\times 3\\ &=27\sin{3x}\\ \end{aligned}\end{equation} $$