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

 RATES

Let

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, &sect;33, we have



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

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,

which is the relation indicated by the above figure.

As a guide in solving rate problems use the following


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


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