Page:Calculus Made Easy.pdf/69



us try the effect of repeating several times over the operation of differentiating a function (see p. 14). Begin with a concrete case.

Let $$y=x^5$$. There is a certain notation, with which we are already acquainted (see p. 15), used by some writers, that is very convenient. This is to employ the general symbol $$f(x)$$ for any function of $$x$$. Here the symbol $$f(\;\;)$$ is read as “function of,” without saying what particular function is meant. So the statement $$y=f(x)$$ merely tells us that $$y$$ is a function of $$x$$, it may be $$x^2$$ or $$ax^n$$, or $$\cos{x}$$ or any other complicated function of $$x$$.

The corresponding symbol for the differential coefficient is $$f'(x)$$, which is simpler to write than $$\frac {dy}{dx}$$. This is called the “derived function” of $$x$$.