Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/165

RATES

94. The derivative considered as the ratio of two rates. Let

$y=f(x)$

be the equation of a curve generated by a moving point P. Its coördinates x and y may then be considered as functions of the time, as explained in § 71. Differentiating with respect to t, by XXV, §33, we have

Plot of curve generated by P.

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

At any instant the time rate of change of y (or the function) equals its derivative multiplied by the time rate change of the independent variable.

Or, write (32) in the form

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

The derivative measures the ratio of the time rate of change of y to that of x.

${\tfrac {ds}{dt}}$ being the time rate of change of length of arc, we have from (12),§71,

(34)	${\frac {ds}{dt}}={\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dt}{dt}}\right)^{2}}}.$

which is the relation indicated by the above figure.

As a guide in solving rate problems use the following rule:

First Step. Draw a figure illustrating the problem. Denote by x, y, z, etc., the quantities which vary with the time.

Second Step. Obtain a relation between the variables involved which will hold true at any instant.