There are several different notations for the derivative of a
function. If $y=f(x)$, the derivative with respect to $x$ may be written as $$f'(x), \quad
\frac{df}{dx}, \quad \frac{dy}{dx}, \quad \hbox{or } D_x
f.$$
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 $$f'', \quad
\frac{d^2 f}{dx^2}, \quad \frac{d^2y}{dx^2}, \quad \hbox{or } D_x^2
f.$$ 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).