One application for derivatives is to estimate an unknown value of a function at a point by using a known value of a function at some given point together with its rate of change at the given point. If $f(x)$ is a function defined on an interval $[a,a+h]$, then the amount of change of $f(x)$ over the interval is the change in the $y$ values of the function over that interval and is given by

The average rate of change of the function $f$ over that same interval is the ratio of the amount of change over that interval to the corresponding change in the $x$ values. It is given by

As we already know, the instantaneous rate of change of $f(x)$ at $a$ is its derivative

For small enough values of $h, \, f^{\prime}(a)\approx \frac{f(a+h)-f(a)}{h}$. We can then solve for $f(a+h)$ to get the amount of change formula:

We can use this formula if we know only $f(a)$ and $f^{\prime}(a)$ and wish to estimate the value of $f(a+h)$. 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 $f(a+h)$ by the $y$ coordinate at $a+h$ on the line tangent to $f(x)$ at $x=a$. Observe that the accuracy of this estimate depends on the value of $h$ as well as the value of $f^{\prime}(a)$.

Figure 1. The new value of a changed quantity equals the original value plus the rate of change times the interval of change: $f(a+h)\approx f(a)+f^{\prime}(a)h$.

Watch the following video to see the worked solution to Example: Estimating the Value of a Function.