Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/52

From (D) we also get:

$\scriptstyle {{\frac {dy}{dx}}={\underset {\Delta x=0}{\operatorname {limit} }}\;{\frac {\Delta y}{\Delta x}}}$.

The process of finding the derivative of a function is called differentiation.

It should be carefully noted that the derivative is the limit of the ratio, not the ratio of the limits. The latter ratio would assume the form $\scriptstyle {\frac {0}{0}}$, which is indeterminate (§14, p12).

29. Symbols for derivatives. Since $\scriptstyle {\Delta y}$ and $\scriptstyle {\Delta x}$ are always finite and have definite values, the expression$\scriptstyle {\frac {\Delta y}{\Delta x}}$is really a fraction. The symbol

$\scriptstyle {\frac {dy}{dx}}$,

however, is to be regarded not as a fraction but as the limiting value of a fraction. In many cases it will be seen that this symbol does possess fractional properties, and later on we shall show how meanings may be attached to $\scriptstyle {dy}$ and $\scriptstyle {dx}$, but for the present the symbol $\scriptstyle {\frac {dy}{dx}}$ is to be considered as a whole.

Since the derivative of a function of $\scriptstyle {x}$ is in general also a function of $\scriptstyle {x}$, the symbol $\scriptstyle {f'(x)}$ is also used to denote the derivative of $\scriptstyle {f(x)}$.

Hence, if	$\scriptstyle {y}$	$\scriptstyle {=f(x)}$,
we may write	$\scriptstyle {\frac {dy}{dx}}$	$\scriptstyle {=f'(x),}$

which is read the derivative of $\scriptstyle {y}$ with respect to $\scriptstyle {x}$ equals $\scriptstyle {f}$ prime of $\scriptstyle {x}$. The symbol$\scriptstyle {\frac {d}{dx}}$when considered by itself is called the differentiating operator, and indicates that any function written after it is to be differentiated with respect to $\scriptstyle {x}$. Thus

$\scriptstyle {\frac {dy}{dx}}$ or $\scriptstyle {{\frac {d}{dx}}y}$ indicates the derivative of $\scriptstyle {y}$ with respect to $\scriptstyle {x}$;
$\scriptstyle {{\frac {d}{dx}}f(x)}$ indicates the derivative of $\scriptstyle {f(x)}$ with respect to $\scriptstyle {x}$;
$\scriptstyle {{\frac {d}{dx}}(2x^{2}+5)}$ indicates the derivative of $\scriptstyle {2x^{2}+5}$ with respect to $\scriptstyle {x}$.
$\scriptstyle {y'}$ is an abbreviated form of $\scriptstyle {\frac {dy}{dx}}$.