- Calculate how quantities change on average over time
- Use rates of change to figure out how an object’s position, speed, and acceleration are changing over time
- Estimate future population sizes using current data and how fast the population is growing
- Use derivatives to determine the cost and revenue of producing one more unit in a business
Amount of Change Formula
One application of derivatives is to estimate an unknown value of a function at a point by using a known value of the function at some given point together with its rate of change at that given point.
If is a function defined on an interval , then the amount of change of over the interval is the change in the values of the function over that interval and is given by:
The average rate of change of the function over that same interval is the ratio of the amount of change over that interval to the corresponding change in the values. It is given by:
As we already know, the instantaneous rate of change of at is its derivative,
For small enough values of . We can then solve for to get the amount of change formula:
We can use this formula if we know only and and wish to estimate the value of . For example, we may use the current population of a city and the rate at which it is growing to estimate its population in the near future.
As we can see in Figure 1, we are approximating by the coordinate at on the line tangent to at . Observe that the accuracy of this estimate depends on the value of as well as the value of .

average rate of change
The average rate of change of a function over the interval is the ratio of the amount of change in over that interval to the corresponding change in values.
The average rate of change is given by:
If and , estimate .