Page:Calculus Made Easy.pdf/69

CHAPTER VII.

SUCCESSIVE DIFFERENTIATION.

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 $y=x^{5}$.

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

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 $f(x)$ 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 $x$.