Page:Calculus Made Easy.pdf/69

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

Let ${\displaystyle y=x^{5}}$.

First differentiation,	${\displaystyle 5x^{4}}$.
Second differentiation,	${\displaystyle 5\times 4x^{3}}$	${\displaystyle =20x^{3}}$.
Third differentiation,	${\displaystyle 5\times 4\times 3x^{2}}$	${\displaystyle =60x^{2}}$.
Fourth differentiation,	${\displaystyle 5\times 4\times 3\times 2x}$	${\displaystyle =120x}$.
Fifth differentiation,	${\displaystyle 5\times 4\times 3\times 2\times 1}$	${\displaystyle =120}$.
Sixth differentiation,		${\displaystyle =0}$.

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 ${\displaystyle f(x)}$ for any function of ${\displaystyle x}$. Here the symbol ${\displaystyle f(\;\;)}$ is read as “function of,” without saying what particular function is meant. So the statement ${\displaystyle y=f(x)}$ merely tells us that ${\displaystyle y}$ is a function of ${\displaystyle x}$, it may be ${\displaystyle x^{2}}$ or ${\displaystyle ax^{n}}$, or ${\displaystyle \cos {x}}$ or any other complicated function of ${\displaystyle x}$.

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