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