Page:Calculus Made Easy.pdf/70

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

Now let us generalize.

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

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

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

if the original function be $y=f(x)$;

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

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

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

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