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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:

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

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:

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

Simplifies to:

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

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!

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

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.