Suppose we differentiate over again, we shall get the “second derived function” or second differential coefficient, which is denoted by ${\displaystyle f''(x)}$; and so on.

Now let us generalize.

Let ${\displaystyle y=f(x)=x^{n}}$.

First differentiation,	${\displaystyle f'(x)=nx^{n-1}}$.
Second differentiation,	${\displaystyle f''(x)=n(n-1)x^{n-2}}$.
Third differentiation,	${\displaystyle f'''(x)=n(n-1)(n-2)x^{n-3}}$.
Fourth differentiation,	${\displaystyle f''''(x)=n(n-1)(n-2)(n-3)x^{n-4}}$.
	etc., etc.

But this is not the only way of indicating successive differentiations. For,

if the original function be ${\displaystyle y=f(x)}$;

once differentiating gives ${\displaystyle {\frac {dy}{dx}}=f'(x)}$;

twice differentiating gives ${\displaystyle {\frac {d({\frac {dy}{dx}})}{dx}}=f''(x)}$;

and this is more conveniently written as ${\displaystyle {\frac {d^{2}y}{(dx)^{2}}}}$, or more usually ${\displaystyle {\frac {d^{2}y}{dx^{2}}}}$. Similarly, we may write as the result of thrice differentiating, ${\displaystyle {\frac {d^{3}y}{dx^{3}}}=f'''(x)}$.