Notation and Higher-Order Derivatives

Let's look at an example to clarify this notation.  Let $y=f(x)=3x^2$.  We will write this derivative as
$$f'(x),\quad y', \quad\frac{dy}{dx},\quad\frac{d}{dx}(3x^2), \text{ or even }(3x^2)'$$

Since the derivative $f'$ is a function in its own right, we can compute the derivative of $f'$. This is called the second derivative of $f$, and is denoted $$f'', \quad y'', \quad \frac{d^2y}{dx^2}, \quad \hbox{or } \frac{d^2 f}{dx^2}.$$ The second derivative tells us how quickly the first derivative is changing, or how quickly the original function is curving (more on that later). We can also compute the third derivative $f'''$ of $f$, which is the derivative of $f''$, or the fourth derivative, which is the derivative of $f'''$. And so on (provided that all those functions are differentiable).

Instead of writing $f'''''(x)$ for the 5th derivative of $f$, we write $f^{(5)}(x)$.  We use this notation when there are too many prime marks to be easily readable.