There are several different notations for the derivative of a
function in this class. If $y=f(x)$, the derivative with respect to
$x$ may be written as $$f'(x), \quad y', \quad \frac{dy}{dx},
\quad\text{or }\quad \frac{df}{dx}.$$ Note:
$$\frac{d(\quad)}{d[\quad ]}$$ means "the derivative of $(\quad )$
with respect to $[\quad ]$.

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.