The paradox of the derivative | Chapter 2, Essence of calculus

The goal here is simple, explain what a derivative is.

The thing is though, there's some subtlety to this topic,

and a lot of potential for paradoxes if you're not careful.

So a secondary goal is that you have an appreciation

for what those paradoxes are and how to avoid them.

You see, it's common for people to say that the derivative measures an instantaneous

rate of change, but when you think about it, that phrase is actually an oxymoron.

Change is something that happens between separate points in time,

and when you blind yourself to all but just a single instant,

there's not really any room for change.

You'll see what I mean more as we get into it,

but when you appreciate that a phrase like instantaneous rate of change is actually

nonsense, I think it makes you appreciate just how clever the fathers of calculus

were in capturing the idea that phrase is meant to evoke,

but with a perfectly sensible piece of math, the derivative.

As our central example, I want you to imagine a car that starts at some point A,

speeds up, and then slows down to a stop at some point B 100 meters away,

and let's say it all happens over the course of 10 seconds.

That's the setup to have in mind as we lay out what the derivative is.

Well, we could graph this motion, letting the vertical axis represent the

distance traveled, and the horizontal axis represent time, so at each time t,

represented with a point somewhere on the horizontal axis,

the height of the graph tells us how far the car has traveled in total after

that amount of time.

It's pretty common to name a distance function like this s of t.

I would use the letter d for distance, but that

guy already has another full time job in calculus.

Initially, the curve is quite shallow, since the car is slow to start.

During that first second, the distance it travels doesn't change that much.

For the next few seconds, as the car speeds up,

the distance traveled in a given second gets larger,

which corresponds to a steeper slope in this graph.

Then towards the end, when it slows down, that curve shallows out again.

If we were to plot the car's velocity in meters per second as a function of time,

it might look like this bump.

At early times, the velocity is very small.

Up to the middle of the journey, the car builds up to some maximum velocity,

covering a relatively large distance each second.

Then it slows back down towards a speed of zero.

These two curves are definitely related to each other.

If you change the specific distance vs.

time function, you'll have some different velocity vs.

time function.

What we want to understand is the specifics of that relationship.

Exactly how does velocity depend on a distance vs.

time function?

To do that, it's worth taking a moment to think

critically about what exactly velocity means here.

Intuitively, we all might know what velocity at a given moment means,

it's just whatever the car's speedometer shows in that moment.

Intuitively, it might make sense that the car's velocity should be higher at times when

this distance function is steeper, when the car traverses more distance per unit time.

But the funny thing is, velocity at a single moment makes no sense.

If I show you a picture of a car, just a snapshot in an instant,

and I ask you how fast it's going, you'd have no way of telling me.

What you'd need are two separate points in time to compare.

That way you can compute whatever the change in distance across those times is,

divided by the change in time.

Right?

I mean, that's what velocity is, it's the distance traveled per unit time.

So how is it that we're looking at a function for velocity that

only takes in a single value of t, a single snapshot in time?

It's weird, isn't it?

We want to associate individual points in time with a velocity,

but actually computing velocity requires comparing two separate points in time.

If that feels strange and paradoxical, good!

You're grappling with the same conflicts that the fathers of calculus did.

And if you want a deep understanding for rates of change, not just for a moving car,

but for all sorts of things in science, you're going to need to resolve this apparent

paradox.

First, I think it's best to talk about the real world,

and then we'll go into a purely mathematical one.

Let's think about what the car's speedometer is probably doing.

At some point, say 3 seconds into the journey,

the speedometer might measure how far the car goes in a very small amount of time,

maybe the distance traveled between 3 seconds and 3.01 seconds.

Then it could compute the speed in meters per second as that tiny

distance traversed in meters divided by that tiny time, 0.01 seconds.