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.
That is, a physical car just side-steps the paradox and
doesn't actually compute speed at a single point in time.
It computes speed during a very small amount of time.
So let's call that difference in time dt, which you might think of as 0.01 seconds,
and let's call that resulting difference in distance ds.
So the velocity at some point in time is ds divided by dt,
the tiny change in distance over the tiny change in time.
Graphically, you can imagine zooming in on some point of this distance vs.
time graph above t equals 3.
That dt is a small step to the right, since time is on the horizontal axis,
and that ds is the resulting change in the height of the graph,
since the vertical axis represents the distance traveled.
So ds divided by dt is something you can think of as the rise
over run slope between two very close points on this graph.
Of course, there's nothing special about the value t equals 3.
We could apply this to any other point in time,
so we consider this expression ds over dt to be a function of t,
something where I can give you a time t and you can give me back the value of this
ratio at that time, the velocity as a function of time.
For example, when I had the computer draw this bump curve here,
the one representing the velocity function, here's what I had the computer actually do.
First, I chose a small value for dt, I think in this case it was 0.01.
Then I had the computer look at a whole bunch of times t between 0 and 10,
and compute the distance function s at t plus dt,
and then subtract off the value of that function at t.
In other words, that's the difference in the distance traveled between the given time,
t, and the time 0.01 seconds after that.
Then you can just divide that difference by the change in time, dt,
and that gives you velocity in meters per second around each point in time.
So with a formula like this, you could give the computer any curve representing any
distance function s of t, and it could figure out the curve representing velocity.
Now would be a good time to pause, reflect, and make sure this idea
of relating distance to velocity by looking at tiny changes makes sense,
because we're going to tackle the paradox of the derivative head on.
This idea of ds over dt, a tiny change in the value of the function s divided by
the tiny change in the input that caused it, that's almost what a derivative is.
And even though a car's speedometer will actually look at a concrete change in time,
like 0.01 seconds, and even though the drawing program here is looking at an actual
concrete change in time, in pure math the derivative is not this ratio ds over dt for a
specific choice of dt. Instead, it's whatever that ratio approaches as your choice for dt
approaches 0.
Luckily there is a really nice visual understanding for what it means to ask what
this ratio approaches, Remember, for any specific choice of dt,
this ratio ds over dt is the slope of a line passing through two separate points
on the graph, right?
Well as dt approaches 0, and as those two points approach each other,
the slope of the line approaches the slope of a line that's
tangent to the graph at whatever point t we're looking at.
So the true honest-to-goodness pure math derivative is not the
rise over run slope between two nearby points on the graph,
it's equal to the slope of a line tangent to the graph at a single point.
Now notice what I'm not saying, I'm not saying that the derivative is
whatever happens when dt is infinitely small, whatever that would mean.
Nor am I saying that you plug in 0 for dt.
This dt is always a finitely small non-zero value, it's just that it approaches 0 is all.
I think that's really clever.
Even though change in an instant makes no sense,
this idea of letting dt approach 0 is a really sneaky backdoor
way to talk reasonably about the rate of change at a single point in time.
Isn't that neat?
It's kind of flirting with the paradox of change in
an instant without ever needing to actually touch it.
And it comes with such a nice visual intuition too,
as the slope of a tangent line to a single point on the graph.
And because change in an instant still makes no sense,
I think it's healthiest for you to think of this slope not as some instantaneous
rate of change, but instead as the best constant approximation for a rate of
change around a point.
By the way, it's worth saying a couple words on notation here.
Throughout this video I've been using dt to refer to a tiny change in t with
some actual size, and ds to refer to the resulting change in s,
which again has an actual size, and this is because that's how I want you to
think about them.
But the convention in calculus is that whenever you're using the letter d like this,
you're kind of announcing your intention that eventually you're
going to see what happens as dt approaches 0.
For example, the honest-to-goodness pure math derivative is written as ds divided by dt,
even though it's technically not a fraction per se,
but whatever that fraction approaches for smaller and smaller nudges in t.
I think a specific example should help here.
You might think that asking about what this ratio approaches
for smaller and smaller values would make it much more difficult to compute,
but weirdly it kind of makes things easier.
Let's say you have a given distance vs time function that happens to be exactly t cubed.
So after 1 second the car has traveled 1 cubed equals 1 meters,
after 2 seconds it's traveled 2 cubed, or 8 meters, and so on.
Now what I'm about to do might seem somewhat complicated,
but once the dust settles it really is simpler,
and more importantly it's the kind of thing you only ever have to do once in calculus.
Let's say you wanted to compute the velocity, ds divided by dt,
at some specific time, like t equals 2.
For right now let's think of dt as having an actual size,
some concrete nudge, we'll let it go to 0 in just a bit.
The tiny change in distance between 2 seconds and 2 plus dt
seconds is s of 2 plus dt minus s of 2, and we divide that by dt.
Since our function is t cubed, that numerator looks like 2 plus dt cubed minus 2 cubed.
And this is something we can work out algebraically.
Again, bear with me, there's a reason I'm showing you the details here.
When you expand that top, what you get is 2 cubed plus 3 times 2 squared dt
plus 3 times 2 times dt squared plus dt cubed, and all of that is minus 2 cubed.
Now there's a lot of terms, and I want you to remember that it looks like a mess,
but it does simplify.
Those 2 cubed terms cancel out.
Everything remaining here has a dt in it, and since there's a dt on the bottom there,
many of those cancel out as well.
What this means is that the ratio ds divided by dt has boiled down into
3 times 2 squared plus 2 different terms that each have a dt in them.
So if we ask what happens as dt approaches 0, representing the idea of looking at a
smaller and smaller change in time, we can just completely ignore those other terms.
By eliminating the need to think about a specific dt,
we've eliminated a lot of the complication in the full expression.
So what we're left with is this nice clean 3 times 2 squared.
You can think of that as meaning that the slope of a line tangent to
the point at t equals 2 of this graph is exactly 3 times 2 squared, or 12.
And of course, there's nothing special about the time t equals 2.
We could more generally say that the derivative
of t cubed as a function of t is 3 times t squared.
Now take a step back, because that's beautiful.
The derivative is this crazy complicated idea.
We've got tiny changes in distance over tiny changes in time,
but instead of looking at any specific one of those,
we're talking about what that thing approaches.
I mean, that's a lot to think about.
And yet what we've come out with is such a simple expression, 3 times t squared.
And in practice, you wouldn't go through all this algebra each time.
Knowing that the derivative of t cubed is 3t squared is one of those things that all
calculus students learn how to do immediately without having to re-derive it each time.
And in the next video, I'm going to show you a nice way to think about
this and a couple other derivative formulas in really nice geometric ways.
But the point I want to make by showing you all of the algebraic guts
here is that when you consider the tiny change in distance caused by a
tiny change in time for some specific value of dt, you'd have kind of a mess.
But when you consider what that ratio approaches as dt approaches 0,
it lets you ignore much of that mess, and it really does simplify the problem.
That right there is kind of the heart of why calculus becomes useful.
Another reason to show you a concrete derivative like this is that it
sets the stage for an example of the kind of paradoxes that come about
if you believe too much in the illusion of instantaneous rate of change.
So think about the actual car traveling according to this t cubed distance function,
and consider its motion at the moment t equals 0, right at the start.
Now ask yourself whether or not the car is moving at that time.
On the one hand, we can compute its speed at that point using the derivative,
3t squared, which for time t equals 0 works out to be 0.
Visually, this means that the tangent line to the graph at that point is perfectly flat,
so the car's quote-unquote instantaneous velocity is 0,
and that suggests that obviously it's not moving.
But on the other hand, if it doesn't start moving at time 0, when does it start moving?
Really, pause and ponder that for a moment.
Is the car moving at time t equals 0?
Do you see the paradox?
The issue is that the question makes no sense.
It references the idea of change in a moment, but that doesn't actually exist.
That's just not what the derivative measures.
What it means for the derivative of a distance function to be 0 is that the best
constant approximation for the car's velocity around that point is 0 m per second.
For example, if you look at an actual change in time,
say between time 0 and 0.1 seconds, the car does move.
It moves 0.001 m.
That's very small, and importantly, it's very small compared to the change in time,
giving an average speed of only 0.01 m per second.
And remember, what it means for the derivative of this motion to be 0 is that
for smaller and smaller nudges in time, this ratio of m per second approaches 0.
But that's not to say that the car is static.
Approximating its movement with a constant velocity of 0 is,
after all, just an approximation.
So whenever you hear people refer to the derivative as an instantaneous rate of change,
a phrase which is intrinsically oxymoronic, I want you to think of that as a
conceptual shorthand for the best constant approximation for rate of change.
In the next couple videos, I'll be talking more about the derivative,
what it looks like in different contexts, how do you actually compute it,
why is it useful, things like that, focusing on visual intuition as always.