Now that we know how to compute the derivatives of many common functions, we can give a few
examples of why the derivative is such a useful tool. In this chapter, we will look at
four different applications of the derivative.
The first application, is to use the derivative to find the velocity and
acceleration of a particle moving in a straight line. When
we are given a function f (t) describing the position of a particle at time t, the velocity of
the particle at time t is the derivative f'(t) and the acceleration is the second
derivate f''(t).
The second application is the analysis of graphs of functions . We can
use the derivative to find critical points and inflection points on graphs, from which a reasonably good
sketch of a function can be constructed.
The second applications is related to the third, optimization of
functions . For example, one may
encounter a function in the business world that gives the total profit of producing a
certain number of goods. It would then be natural to try to maximize such a function.
The fourth and final application concerns related rates . Suppose water
is flowing into a gigantic ice cream cone at a fixed rate (for some strange reason). Through a clever
application of differentiation, it is possible to determine how quickly the water level will be rising when it reaches any
particular height in the cone.
