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.