Suppose we wanted to measure a runner’s instantaneous speed using a stopwatch. By instantaneous speed, we mean their speed at an exact moment in time.

Let’s define *f(t)* as the runner’s distance from the start time at time *t*. How could we find the runner’s instantaneous speed at 1 second after the race? We could record their positions at 1 second and some time after 1 second and then do some calculations. Let’s say we measure their distance from the start at *f(1)* and *f(3)*. The change in position is *f(3) - f(1)*, and the time between 1 second and 3 seconds is 2 seconds, so we can calculate the runner’s average speed to be the following:

`$average\ speed = \frac{f(3)-f(1)}{3-1}$`

However, this gives us the average speed, not the instantaneous speed. We don’t know the runner’s speed at *t=1*; we only know their speed on average over the 2-second interval.

If we repeated the process but instead took our second measurement at 1.1 seconds, we could be more accurate since we would find the average speed between 1 and 1.1 seconds. If we took our second measurement at a tiny increment, such as *t=1.0000001*, we would be **approaching** instantaneous speed and get a very accurate measurement. The animation to the right demonstrates this concept.

We can generalize this using limits. Define *t = 1* to be the first measurement time, and let’s say we wait *h* seconds until taking the second measurement. Then the second time is *x+h*, and the positions at the two times are *f(x)* and *f(x+h)*. Repeating the exact process as above, we define the runner’s average speed as:

`$average\ speed = \frac{f(x+h)-f(x)}{(x+h)-x}$`

Simplifies to:

`$average\ speed = \frac{f(x+h)-f(x)}{h}$`

Using limits, we can make *h* very small. By taking the limit as *h* goes to 0, we can find the instantaneous rate of change!

`$instantaneous\ rate\ of\ change\ = \lim_{h \rightarrow 0} \frac{f(x+h)}{h}$`

This is called the *derivative at a point*, which is the function’s slope (rate of change) at a specific point. In the next few exercises, we will dive further into derivatives.

**Note**: We have shown examples where *h* is positive, but it can also be a negative value approaching 0.