# The paradox of the derivative | Essence of calculus, chapter 2

The goal here is simple: Explain what a derivative

is.

Thing is, though, there’s some subtlely

to this topic, and some potential for paradoxes

if you’re not careful, so the secondary

goal is that you have some appreciation for

what those paradoxes are and how to avoid

them.

You see, it’s common for people to say that

the derivative measures “instantaneous rate

of change”, but if 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 a single instant, there is no more room

for change.

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

and when you appreciate that a phrase like

“instantaneous rate of change” is nonsensical,

it makes you appreciate how clever the fathers

of calculus were in capturing the idea this

phrase is meant to evoke with a perfectly

sensible piece of math: The derivative.

As our central example, imagine a car that

starts at some point A, speeds up, then slows

to a stop at some point B 100 meters away,

all over the course of 10 seconds.

This is the setup I want you to keep in mind

while I lay out what exactly a derivative

is.

We could graph this motion, letting a vertical

axis represent the distance traveled, and

a horizontal axis represent time.

At each time t, represented with a point on

the horizontal axis, the height of the graph

tells us how far the car has traveled after

that amount of time.

It’s common to name a distance function

like this s(t).

I’d use the letter d for distance, except

that it already has another full time job

in calculus.

Initially this curve is quite shallow, since

the car is slow at the start.

During the first second, the distance traveled

by the car hardly changes at all.

For the next few seconds, as the car speeds

up, the distance traveled in a given second

gets larger, corresponding to a steeper slope

in the graph.

And as it slows towards the end, the 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 time t=0, the velocity is 0.

Up to the middle of the journey, the car builds

up to some maximum velocity, covering a relatively

large distance in each second.

Then it slows back down to a speed of 0 meters

per second.

These two curves are highly related to each

other; if you change the specific distance

vs. time function, you’ll have some different

velocity vs. time function.

We want to understand the specifics of this

relationship.

Exactly how does velocity depend on this distance

vs. time function.

It’s worth taking a moment to think critically

about what velocity actually means here.

Intuitively, we all know what velocity at

a given moment means, it’s whatever the

car’s speedometer shows in that moment.

And intuitively, it might make sense that

velocity should be higher at times when the

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, a snapshot

in an instant, and ask you how fast it’s

going, you’d have no way of telling me.

What you need are two points in time to compare,

perhaps comparing the distance traveled after

4 seconds to the distance traveled after 5

second.

That way, you can take the change in distance

over the change in time.

Right?

That’s what velocity is, the distance traveled

over a given amount of time.

So how is it that we’re looking at a function

for velocity that only takes in a single value

for t, a single snapshot in time.

It’s weird, isn’t it?

We want to associate each individual point

in time with a velocity, but computing velocity

requires comparing two points in time.

If that feels strange and paradoxical, good!

You’re grappling with the same conflict

that the fathers of calculus did, and if you

want a deep understanding of rates of change,

not just for a moving car, but for all sorts

of scenarios in science, you’ll need a resolution

to this apparent paradox.

First let’s talk about the real world, then

we’ll go into a purely mathematical one.

Think about what an actual car’s speedometer

might be 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, perhaps

the distance traveled between 3 seconds and

3.01 seconds.

Then it would compute the speed in meters

per second as that tiny distance, in meters,

divided by that tiny time, 0.01 seconds.

That is, a physical car can sidestep the paradox

by not actually computing speed at a single

point in time, and instead computing speed

during very small amounts of time.

Let’s call that difference in time “dt”,

which you might think of as 0.01 seconds,

and call the resulting difference in distance

traveled “ds”.

So the velocity at that point in time is ds

over dt, the tiny change in distance over

the tiny change in time.

Graphically, imagine zooming in on the point

of the distance vs. time graph above t=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

distance traveled.

So ds/dt is the rise-over-run slope between

two very close points on the graph.

Of course, there’s nothing special about

the value t=3, we could apply this to any

other point in time, so we consider this expression

ds/dt to be a function of t, something where

I can give you some time t, and you can give

back to me the value of this ratio at that

time; the velocity as a function of time.

So for example, when I had the computer draw

this bump curve here representing the velocity

function, the one you can think of as the

slope of this distance vs. time function at

each point, here’s what I had computer do:

First, I chose some small value for dt, like

0.01.

Then, I had the computer look at many times

t between 0 and 10, and compute the distance

function s at (t + dt), minus the value of

this function at t.

That is, the difference in the distance traveled

between the given time t, and the time 0.01

seconds after that.

Then divide that difference by the change

in time dt, and this gives the velocity in

meters per second around each point in time.

With this formula, you can give the computer

any curve representing the distance function

s(t), and it can figure out the curve representing

the velocity v(t).

So now would be a good time to pause, reflect,

make sure this idea of relating distance to

velocity by looking at tiny changes in time

dt makes sense, because now we’re going

to tackle the paradox of the derivative head-on.

This idea of ds/dt, a tiny change in the value

of the function s divided by a tiny change

in the input t, is almost what the derivative

is.

Even though out car’s speedometer will look

at an actual change in time like 0.01 seconds

to compute speed, and even though my program

here for finding a velocity function given

a position function also uses a concrete value

of dt, in pure math, the derivative is not

this ratio ds/dt for any specific choice of

dt.

It is whatever value that ratio approaches

as the choice for dt approaches 0

Visually, asking what this ratio approaches

has really a nice meaning: For any specific

choice of dt, this ratio ds/dt is the slope

of a line passing through two points on the

graph, right?

Well, as dt approaches 0, and those two points

approach each other, the slope of that line

approaches the slope of a line tangent to

the graph at whatever point t we’re looking

at.

So the true, honest to goodness 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.

Notice what I’m not saying: I’m not saying

that the derivative is whatever happens when

dt is infinitely small, nor am I saying that

you plug in 0 for dt.

This dt is always a finitely small, nonzero

value, it’s just approaching 0 is all.

So even though change in an instant makes

no sense, this idea of letting dt approach

0 is a really clever backdoor way to talk

reasonably about the rate of change at a single

point in time.

Isn’t that neat?

It’s flirting with the paradox of change

in an instant without ever needing to touch

it.

And it comes with such a nice visual intuition

as the slope of a tangent line at a single

point on this graph.

Since 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 as the best constant

approximation for rate of change around a

point.

It’s worth saying a few 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

tiny change in s, which again has an actual

size.

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 announcing that the intention is

to eventually see what happens as dt approaches

0.

For example, the honest-to-goodness derivative

of the function s(t) is written as ds/dt,

even though the derivative is not a fraction,

per se, but whatever that fraction approaches

for smaller and smaller nudges in t.

A specific example should help here.

You might think that asking about what this

ratio approaches for smaller and smaller values

of dt would make it much more difficult to

compute, but strangely it actually makes things

easier.

Let’s say a given distance vs. time function

was exactly t3.

So after 1 second, the car has traveled 13=1

meters, after 2 seconds, it’s traveled 23=8

meters, and so on.

What I’m about to do might seem somewhat

complicated, but once the dust settles it

really is simpler, and it’s the kind of

thing you only ever have to do once in calculus.

Let’s say you want the velocity, ds/dt,

at a specific time, like t=2.

And for now, think of dt having an actual

size; we’ll let it go to 0 in just a bit.

The tiny change in distance between 2 seconds

and 2+dt seconds is s(2+dt)-s(2), and we divide

by dt.

Since s(t)=t3, that numerator is (2+dt)3

– 23.

Now this, we can work out algebraically.

And again bear with me, there’s a reason

I’m showing you the details.

Expanding the top gives 23 + 3*22dt + 3*2*(dt)2

+ (dt)3 – 23.

There are several terms here, and I want you

to remember that it looks like a mess, but

it simplifies.

Those 23 terms cancel out.

Everything remaining has a dt, so we can divide

that out.

So the ratio ds/dt has boiled down to 3*22

+ two different terms that each have a dt

in them.

So as dt approaches 0, representing the idea

of looking at smaller and smaller changes

in time, we can ignore those!

By eliminating the need to think of a specific

dt, we’ve eliminated much of the complication

in this expression!

So what we’re left with is a nice clean

3*22.

This means the slope of a line tangent to

the point at t=2 on the graph of t3 is exactly

3*22, or 12.

Of course, there was nothing special about

choosing t=2; more generally we’d say the

derivative of t3, as a function of t, is 3*t2.

That’s beautiful.

This derivative is a crazy complicated idea:

We’ve got tiny changes in distance over

tiny changes in time, but instead of looking

at any specific tiny change in time we start

talking about what this thing approaches.

I mean, it’s a lot to think about.

Yet we’ve come out with such a simple expression:

3t2.

In practice, you would not go through all

that algebra each time.

Knowing that the derivative t3 is 3t2 is one

of those things all calculus students learn

to do immediately without rederiving each

time.

And in the next video, I’ll show ways to

think about this and many other derivative

formulas in nice geometric ways.

The point I want to make by showing you the

guts here is that when you consider the change

in distance of a change in time for any specific

value of dt, you’d have a whole mess of

algebra riding along.

But by considering what this ratio approaches

as dt approaches 0, it lets you ignore much

of that mess, and actually simplifies the

problem.

Another reason I wanted to show you a concrete

derivative like this is that it gives a good

example for the kind of paradox that come

about when you believe in the illusion of

an instantaneous rate of change.

Think about this car traveling according to

this t3 distance function, and consider its

motion at moment t=0.

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 of this

function, 3t2, which is 0 at time t=0.

Visually, this means the tangent line to the

graph at that point is perfectly flat, so

the car’s quote unquote “instantaneous

velocity” is 0, which suggests 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 this for a moment,

is that car moving at t=0?

Do you see the paradox?

The issue is that the question makes no sense,

it references the idea of of change in a moment,

which doesn’t exist.

And that’s just not what the derivative

measures.

What it means for the derivative of the distance

function to be 0 at this point is that the

best constant approximation for the car’s

velocity around that point is 0 meters per

second.

For example, between t=0 and t=0.1 seconds,

the car does move… it moves 0.001 meters.

That’s very small, and importantly it’s

very small compared to the change in time,

an average speed of only 0.01 meters per second.

What it means for the derivative of this motion

to be 0 is that for smaller and smaller nudges

in time, this ratio of change in distance

over change in time approaches 0, though in

this case it never actually hits it.

But that’s not to say the car is static.

Approximating its movement with a constant

velocity of 0, after all, just an approximation.

So if you ever hear someone refer to the derivative

as an “instantaneous rate of change”,

a phrase which is intrinsically oxymoronic,

think of it as a conceptual shorthand for

“the best constant approximation for the

rate of change”

In the following videos I’ll talk more about

the derivative; what does it look like in

different contexts, how do you actually compute

it, what’s it useful for, things like that,

focussing on visual intuition as always.

As I mentioned last video, this channel is

largely supported by the community through

Patreon, where you can get early access to

future series like this as I work on them.

One other supporter of the series, who I’m

incredibly proud to feature here, is the Art

of Problem Solving.

Interestingly enough, I was first introduced

to the Art of Problem Solving by my high school

calculus teacher.

It was the kind of relationship where I’d

frequently stick around a bit after school

to just chat with him about math.

He was thoughtful and encouraging, and he

once gave me a book that really had an influence

on me back then.

It showed a beauty in math that you don’t

see in school.

The name of that book?

The Art of Problem Solving.

Fast-forward to today, where the Art of Problems

Solving website offers many many phenomenal

resources for curious students looking to

get into math, most notably their full courses.

This ranges from their newest inspiring offering

to get very young students engaged with genuine

problem solving, called Beast Academy, up

to higher level offerings that cover the kind

of topics that all math curious students should

engage with, like combinatorics, but which

very few school include in their curriculum.

Put simply, they’re one of the best math

education companies I know, and I’m proud

to have them support this series.

You can see what they have to offer by following

the link in the screen, also copied in the

video description.

There's no way a sentence could ever express my appreciation towards you. Your videos dive so deep into these kind of concepts and teach them in a way that no one would have ever thought of. Thank you very much, I honestly can't think of how my math and logic life would ever live on without your teachings so I pray that you keep this up. Thank you once again, Grant. Thank you enormously!

13:52 I feel like this sums up the video pretty well

really really awesome…….nice and intutive….can u make some videos on machine learning?

After watching these videos I looked up the definition of calculus to understand the bigger concept better and I was like holy cow calculus could not be explained better than these videos. Thank you so much!!

For reference-Calculus: the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.

12:25

maffgasm'd

You made it very easy. Thanks a lot.

Is it just luck that dt cancels out in the formula you mention, or is there some "deeper" reason behind it?

You call it beautiful, but it also just feels like it happens randomly.

I am using your videos to learn calculus, so there's a pretty good chance this thought is wrong since I dont know that much yet, lol. Anyway, here it is: the phrase "instantaneous rate of change", kinda made sense to me, I'd just recently gone through a bunch of videos teaching the concepts of general relativity. That's actually why I came here, I wanted to be able to build the skills to understand einstein's equations. Through the process of learning about relativity, I also learned about time as the 4th dimension, and thinking of space as spacetime. With that in mind, would the instantaneous rate of change not describe a property that could be displayed in a 4d representation, where the point is in question is not a 1d point, but a 2d relationship represented as a 1d point? Perhaps the derivative could be the rule that shows the relationship between 2 dimensions, allowing us to show and display the relationship between 3d, and 4d, on a graph? Also, I listened to the introduction, paused the video and wrote this comment, just because I was excited about it, so if you address that later in the video, keep in mind I will watch that part lol. I just had to get that thought off my chest. Anyway, love the videos, I've watched tons of your other ones.

I think dt should be the Planck time, since that's the smallest time step that actually exists

i like watching these videos to make me feel smart while I actually have no idea what he's talking about

Love and respect ❤❤❤

When did π get its driving license ?

This channel is ethereal. You have an incredible talent for intuitive explanations of well known mathematical concepts and your voice is awesome.

Says go into the real world, and then completely boffs how a speedometer actually works to make his point. A speedometer doesn't differentiate position. It measures speed directly like an accelerometer measures acceleration directly. From that perspective, you can talk about an instantaneous rate of change.

Wow, all these mathematicians making fun of engineers for approximating stuff, when they've been doing it all along!

Just as you can take a snapshot of a moving car, you can also take a snapshot of its speedometer, and that's what the car's instantaneous rate of change of its distance is with respect to time. There is no paradox here….

Anyone can be intelligent, but only those who can successfully articulate their incredibly intelligent ideas are brilliant. Great video 👏

I see the fundamental theorem of calculus in this series, the idea is anti derivative is understandable. After questioning myself the idea formation of derivative, I am back to this episode. The idea ‘constant approximation ‘ is really shocking to me.

my low IQ brain lost at the point when he said that rest of the terms in equation can be ignored

Comment section for those who came late.

How do you define a tangent to a graph without using the rise/run approximation dealing with ever decreasing differences? I feel like using the term tangent adds nothing to an understanding.

I've always hugely appreciate how interconnected and complementary the Fundamental Theorem and the definitions of derivative and integral are. Trying to make sense of any one of them without the others is almost meaningless

That einstein quote went over my head anyone care to explain?

I'm korean high school student so.. I cant speak Engkish well

It is difficult to express in English. If you take 0.1 0.001 0.00 … 1 to take a picture and the car leaves afterimages for that 0.00..1 second to take a picture, The instantaneous speed(ds/dt) will make it easier to think of it as the distance / time the picture was taken.

Good day, the site u've recommended doesn't work for me, is it just me or there are some technical problems?

You almost brought me to tears. Thank you for existing and thank you for making this.

I wonder if, for extra mental prep material, Vsauce's videos on SuperTasks would be helpful as it gives a more visual understanding of "dt approaching 0" means and the paradox you're talking about.

Could someone correct me if I am wrong on this, it seems intuitively like it'd be helpful but I do not want to mislead or confuse anyone trying to gain a better understanding of the material presented in the video.

Everything seems like an asymptote or whatever, buncha tricky dick stuff. Its 2:00 am i need to sleep idk what im talking about

Pause and ponder

4:34

"…..and then we'll go into a

BOOMPmathematical one."I feel like somebody just facedesked from not being able to understand anything up to that point.

Thank u so much 3 BLUE 1 BROWN to give maths its real beauty.

Its really a dream come true to see and understand mathematics in such a way.!!!

"You don't see this in school".

I don't know why, it's not the first time I'm seeing this video, but somehow this sentence hit me hard this time.

Why can't they change their education system for the best ? I really can't understand this. If you just go to Youtube, there are today tons upon tons of intuitive approaches to things like Maths that often don't make sense to students. And that is not only true in America (I assume it's the first example whenever someone complains about it), it's true in Europe too (I'm french). They keep their old system that is not adapted to the 21st century anymore, now that we have technologies that let us be able to visualize such things so intuitively.

I'm in my first year of university, and so far so long, I have the feeling that my entire semester in math's class will consist not on solving interesting problems, but just on memorizing a bunch of formulas and stereotypical-problem-solving recipes. Why are they doing this ?

Every possible education system should keep that in mind :

you learn by making experiences, not by memorizing things.If I know how to write this comment in english, it is not because I memorized every possible english grammar rule in school and that I answered every question they gave me correctly, it's because I was practicing it on the internet for a relatively long time. Of course, you won't be able to experience anything if you don't have any bases to begin with, and the problems of education are a bit more complicated than just lack of experience, but you get my main point.

Totally Aristotelian time understanding!

iam not getting the tranformation of distance vs time graph to velocity vs time when distance vs time is derived like it gives slope and single value that is located on the graph itself but the velocity vs time graph is different which comes after differentiation of distance vs time can you explain that.

Very informative and beautiful approach to understand maths

speed of light is a non continues function and constant by nature so it's must be a modules equation of some functions because it's have no acceleration or retardation😝😝😝😝😪😓😌🙌🙈😿👹💀👣👍👎👆👇👈👉⌚🔰💹📓📑📑📒📓📕📖📰📚📙📘📗📔🔭🔬🔮🔦

Hi this is a newb question,pre calc 11 student here going towards pre calc 12,what happened on @6:44 ?Where and how did he get the ds/dt(t) = s(t+dt)-s(t)?I probably missed something about how he came up with that on the right side of equation.

did he just refactor or expanded the ds to make it equal to s(t+dt)-s(t) as it is equal to ds when i investigated a bit.and later on he placed dt in the denominator to make it ds/dt and is identical to the left side of equation.

Am I the only one who has this in middle school?

0-4 kindergarten

4-6 junior infants

6-12 primary school

12-18 secondary school

18+ Collage/University

And I am in secondary school

Beautiful video

There's a duality here between knowing the value of s(t) for all values of t, and knowing the value of all derivatives of s(t) at only one value of t. The "contradiction" in "instantaneous rate of change" only exists if you try to cross those two worlds. "Instantaneous" doesn't refer to s(t) not existing for any other value of t than the one you're looking at, it refers to evaluating the first derivative at exactly one point.

When I took calculus in high school I was extremely fascinated when I first learned about limits and eventually derivatives. I was amazed to be able to find the slope at a single point because using the limit involved dividing by a difference of zero which is impossible, so I was very confused as to how such a problem could be solved. I was very fascinated to learn that it is indeed possible. It's one of the most brilliant discoveries of humanity if you ask me. It's just truly beautiful.

This is the best video in the history of distance over time!!

Um, maybe make up your mind. Are you going to approach the problem head-on? Or give us a sneaky back-door way of looking at it? Those two are not any paradox: they're a contradiction.

On the other hand, it's difficult to see why Bishop Berkley should have objected. He was in the business of forgiving sinners, wasn't he?

1:20 aperture science wheels on a car o_o

What is the paradox?

i dont know who u are, where are you or neither what u are……. u gained my respect……it feels like i will find u and kiss you…then give me a gun……i will lynch my school as well as my tution teacher

Great video, but I have to disagree. Instantaneous rate of change is not paradoxical/oxymoron. It is the slope (of the tangent) of a function at a single point. It is an instantaneous condition, just like an innitial condition.

Did any of you understand why we not put dt infinetly small, but instead let it appriach 0? (8:41)

Maybe somebody could help me out, the only place I got lost was 11:56 when he expands the top line, if I expanded that I'd probably get 2 cubed + dt cubed – dt cubed. I'm obviously missing something basic, but could somebody explain it step by step or tell me what to google?

That MOMENTUM Tab on your Google Chrome..Loved it..I too have it.

I'm a little confused, around 9:00 minutes you tangent line seems to touch the graph at multiple points, doesn't tha contradict the definition of a tangent?

i understand now

Hi! So I am new to calculus like most, I haven't really taken any kind of precalc class before so I really have no prior knowledge of this but was curious. At 7:18, you have the equation s(t+dt)-s(t)/dt and I was curious where that came from

Either you are confusing yourself or you don't understand what you are talking about..

Instantaneous (adj) doesn't mean one point and change (noun) doesn't mean multiple points in time.

You are saying that because you are not making a difference between the N-dimensional space you are working in and successive points (positions) taken by the moving object at INSTANT (t) that means Nx(i) occupied at INSTANT t(i)

but a rate is a limit when dt approaches zero it's in very smal interval of time

Instantaneous velocity is just a limit of a certain (rate of change)

So I recommend that you go back to your high school math and try to review the definition of a derivative of a function which is just a limit of this function in respect to a certain variable Xi when this latter tend to 0

The word "instantaneous " just means "at that INSTANT of time "

An instant of time does not any length of time.

2000th comment

I learned Calculus at age 16 (in 1982).

But I clicked on this video anyway…because they are so well done.

[What follows is meant to be humours, just so you know.]

Before I clicked on this video (and for the past 37 years) I would have said the essence of Calculus is…well something like what was said in the video.

But now, thinking about it again, I think I'd rather say: "Calculus' superpower is Luck"!

Because for the derivative and integral we start off by creating an endless set of problems from pieces of what was a single problem, and then!…because Luck is our superpower (like Zazie Beetz in deadpool 2), that endless set of problems turns out to be easier than the one problem we started with!!

So the best approximation of a derivative woudl be something like ds / dt where dt = planck time ?

The idea that instantaneous velocity "makes no sense" doesn't make sense to me. Just because a car's positional velocity calculation might require 2 positions and 2 separate times to calculate, doesn't mean you couldn't take an instantaneous velocity measurement by some other means that doesn't require separate times…. like a pitot tube! The equation for fluid velocity in a pitot tube requires only the measurement of height of a fluid at a given instant and doesn't depend on time. There is still an absolute instantaneous velocity even though one method of calculating it might require differences in time.

I use derivatives in school for about a year now and it just got explained to use on how it works and not why. We also used a term similar to "Instantaneous rate of change" to describe it. Also we just were given the formula to calculate the derivative with. At 12:30 in this video my brain had the biggest click moment it ever had in maths. You literally could feel the light bulb over my head turning on and I thought to myself "Why tf aren't they teaching us this beauty?" I feel like that's one of the most beautiful, yet so easy to come up with things I ever saw in maths.

I m not expert in math but your videos gives me a new vision on the same things. Greetings from Slovakia. 🙂

I love maths and the paradox felt very ez

I still haven't studied this at school, but it's pretty fun

6:39 he lost me right here

This is genius

The way I was taught I always new that when we use the derivation formula I can get the derivation formula in the table but to see it done so cheeky and amazing

Maybe it’s been asked before: But what is the technical difference between „infinitely small“ and a fixed distance that approaches 0?

Just realized the channel name refers to the 3 pi students and the single teacher.

Think about this paradox: as the space of real numbers has the "power of c o n t i n u u m" property, meaning you can make an arbitrarily small infinitesimal change on the number axis without it being zero, i.e. you can approach a number arbitrarily near, and the function f(x) is continuous – how come there is a UNIQUE value of the limit of the slope value at all ? 🤔😏… It is some sort of a mathematical idealization. I guess, just like the continuous functions are!🍺

The way I always think about it is instantaneous rate of change would be the rate of change if it could just be by itself, so for the car example the instantaneous rate of change would be the rate of change if all other forces stopped acting on the car, like where would it go?

So at t=0 if all other forces stopped affecting the car then it will just sit there.

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I think of the velocity of the car at an instant as what the velocity would be if the car continued doing what it was doing at that point.In other words what would the velocity be if all forces acting on the car were removed.If forces forces were removed then the car just do what it was doing at that particular point in time.

"infinitely small" works too. I think (personally) that it's harder (what exactly does "infinitely small" mean?) to do the analysis (theory of why calculus works), but engineers and scientists never stopped using "infinitely small" after Cauchy told us it was not logically sound. Then Abraham Robinson proved it is. (1957? 1964?)

The only (modern) calculus book I've seen with infinitesimals is Keisler's _Calculus_. He does not even attempt to justify using infinitesimals; rather he gives a couple of axioms and off we go. And he justifies the usual approximations of the derivative (the things you take a limit of) as "approximations". I would not have bought this when I took calculus. (I hadn't yet taken Logic and proved the Compactness Theorem, from which the existence of infinitesimals in some model of the reals [typically called "the" nonstandard reals] follows. And being somewhat [although not completely] of an Constructivist, I don't buy "take any nonprincipal ultrafilter over the reals" as a construction. Maybe I need more experience with ultrafilters.)

I think all of us in the comment section want to thank your supporters on Patreon for making these videos possible for us. Thank you.

newton discovered nothing, he just plagiarized Leibniz and only in one book and no more other work he has done if one can call copying a work

I know it’s a bit late, but may I just say that this is an absolutely phenomenal series. It really helped me understand calculus in a way that just didn’t happen at school.

WRONGThe car at time 0 has the instantaneous rate of change of 0.

Your inability to wrap your mind correctly around this idea caused you to redefine it as the best approximation, which is wrong. It's not an approximation. It's the EXACT instantaneous velocity.

This might help you get the point. A car is standing still at a light. The light turns green. The driver hits the gas. The car moves according to your graph.

Any point in time can arbitrarily set as 0. What we choose to do is this:

The time the car is moving is to the right of the axis.

The time the car is not moving is to the left of the axis.

What about 0?

That is the last point where it's not moving.

Not moving = [-? , 0]

Moving = (0,?)

Do you remember ( vs [ in math. ( ) means the range ends at those points but does not include the points. [ ] means the range ends at those points and includes the points.

Thus, if the car starts at time 0 and stops at time 10, you are imagining the range of the car's moving to be [0,10] when it's actually (0,10). This small difference in perception has caused you to think they're all wrong and your definition is right.

I did not get where the (3(2)^2dt +3(2)(dt)^2+(dt)^3) in the numerator came from

Now I understand

Really the best explanation about calculus

Thankyou very much sir

We expect more of such videos

I m really fascinated by ur videos ,,,thanks for making me understand nature so well

at exactly 12:34 I was like "you m*****er"

this is truely wholsome explanation

why is dt not infinitely small and instead finitely small?

You give me hope in my life and spark excitement, no joke.

You’ve illustrated speed, not velocity, as there is no direction.

My favourite derisive example will always be relating derivatives to the speed that a speedometer reads

@3:34 derivatives are functions with symbols. If the picture comes with a number giving it's speed then it's not a paradox.

Can we just stop for a second and appreciate the little Pi driving the car?

Not sure if I agree with the last example, I don't think it's a paradox to say the car isn't moving. The car has 0 velocity and 0 acceleration but it does have a positive jerk (third derivative) of 6, which will increase acceleration and velocity eventually causing the car to move.

The "when does the car start moving" paradox reminds me of the "when does the arrow hit the runner" paradox. Both deal with movement and both are impossible to solve if you keep calculating on and on. Until you say: that last term is approximately zero.

If there was a Nobel prize for teaching this guy would be it.

This is absolutely beautiful.

This felt like it was more geared towards an audience that already knows the subject, but for a beginner, you would start linear function, m=Δy/Δx and then bringing them to the idea of dy/dx for an instantaneous rate of change, using the difference quotient definition.

can I ask what the term "Nonzero size… for now", mean in 14:02? thank you in advance

imo it would've been better if you reffered to it as f(x) and f(x+h)

dy/dx = (f(x+h) -f(x))/h

This whole video is a "d(OMG)/dt" moment!

BECAUSE HUMANS ARE NOT PERFECT OUR SYSTEMS ARE NOT PERFECT AS GOD SAYS IT SO. STILL ONE CAN MAKE IT TO WHERE THE MARGION OF ERROR IS SO SMALL THAT IT IS NOT SEEN.

What happens if you make dt and ds

a square?

a formula for derivitave of f(n) could be

lim (f(n+x)-f(n))/x

x->0+

so

d(f(n)) = f(n+x)-f(n)

🙂

btw, dx = 0+

These series of videos gave me a sense of what calculus is? I was searching for the answers for the last 20 years and I got it now. Thanks a lot.

How to represent f(x)=x^2 in graph, in simillar to f(x)=1/x by considering area concept?

What a generous contribution to the world – thank you so much for these videos