Introduction

Everyone has an intuitive sense of speed as "distance covered per unit time," but that intuition is really about average speed. If a bus covers $60$ km in one hour, its average velocity for the trip is $60$ km/h — yet it plainly slows in towns and speeds up on open stretches, so the actual velocity keeps changing. Knowing the average tells you nothing about the velocity at one particular instant.

In general, $v_{\text{ave}} = \dfrac{\text{distance travelled}}{\text{time of travel}}$. A runner who covers $10$ km in $1.25$ h has $v_{\text{ave}} = 10/1.25 = 8$ km/h for the whole race. But suppose we want the runner's velocity at the exact instant halfway through: if the distance covered in $[0\text{ h}, 0.5\text{ h}]$ is $5$ km, the average velocity over that half is $v_{\text{ave}} = 5/0.5 = 10$ km/h — a different (and still only average) number. Shrinking the interval further, from $0$ h to $0.6$ h the runner has covered $5.7$ km, so

for the interval $[0.5,0.6]$ — a better estimate of the instantaneous rate at $t=0.5$ h than the $8$ km/h race-average was, precisely because the time window is shorter. "Shrinking" the interval between $0.5$ h and a time very close to it keeps producing better and better approximations to the true velocity at that one instant. (For comparison, Usain Bolt's $100$ m world-record run, $9.58$ s, gives an average speed of $100/9.58 \approx 10.44$ m/s — but the question "how fast is Bolt running at this exact instant?" needs the same shrinking-interval idea, not just the race average.)

This problem of finding an instantaneous rate from a general functional relationship $y=f(x)$ is exactly the problem calculus was invented to solve. As Leibnitz's epigraph for this chapter puts it: "Take what you need, do what you should, you will get what you want." What we need is a way to let the length of the time (or, more generally, the $x$-) interval shrink all the way to zero while still getting a meaningful, finite number out the other end. That number is the derivative, and building the tools to compute it — and to interpret it geometrically as a slope and physically as a rate — is the business of this chapter.