Normal tangents, Rate measure, approximation and error.

APPLICATION OF DERIVATIVES
TANGENTS AND NORMALS,
RATE MEASURE

,DERIVATIVE AS THE RATE OF CHANGE:
or derivative as the rate of change
If a variable quantity is some function of time i.e., , then small change in tim
corresponding in .
Thus the average rate of change
When limit is applied, the rate of change becomes instantaneous and we get th
change of w.r.t. time at an instant.
i.e.,
Hence it is clear that the rate of change of any variable with respect to some other va
derivative of first variable with respect to other variable.

,Note:
[i] The value of at i.e. represent the rate of change of with respect to at .
[ii] If and , then , provided that .
[iii] Throughout this chapter the term "rate of change" will mean the instantaneou
change unless stated otherwise.

, Illustrating the Concept:
If the radius of a circle is increasing at a uniform rate of 2 cm/sec. Find the
increasing of area of circle, at the instant when the radius is 20 cm.
Given [where radius and time]
Now, area of circle is given by


Thus, rate of change of area of circle with respect to time is