Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/155

DIFFERENTIALS

87. Introduction. Thus far we have represented the derivative of $y=f(x)$ by the notation

${\frac {dy}{dx}}=f'(x).$

We have taken special pains to impress on the student that the symbol

${\frac {dy}{dx}}$

was to be considered not as an ordinary fraction with dy as numerator and dx as denominator, but as a single symbol denoting the limit of the quotient

${\frac {\Delta y}{\Delta x}}$

as $\Delta x$ approaches the limit zero.

Problems do occur, however, where it is very convenient to be able to give a meaning to dx and dy separately, and it is especially useful in applications of the Integral Calculus. How this may be done is explained in what follows.

88. Definitions. If $f'(x)$ is the derivative of $f(x)$ for a particular value of x, and $\Delta x$ is an arbitrarily chosen increment of x, then the differential of $f(x)$, denoted by the symbol $df(x)$, is defined by the equation

(A)	$df(x)=f'(x)\Delta x.$

If now $f(x)=x$, then $f'(x)=1$, and (A) reduces to

$dx=\Delta x,$

showing that when x is the independent variable, the differential of x (= dx) is identical with $\Delta x$. Hence, if $y=f(x)$, (A) may in general be written in the form

(B)	$dy=f'(x)dx.$[1]

1. On account of the position which the derivative $f'(x)$ here occupies, it is sometimes called the differential coefficient.

   The student should observe the important fact that, since dx may be given any arbitrary value whatever, dx is independent of x. Hence, dy is a function of two independent variables x and dx.