Next up is integrals. Follow the full playlist at 3b1b.co/calculus By the way, there is a piece of math, commonly called "non-standard analysis", which makes infinitesimals a rigorous notion, thereby avoiding the need to use limits. That is, in the real number system something like 0.000....(infinitely many 0's)...1 doesn't make sense, it's not an actual number. But the "hyperreal numbers" of non-standard analysis are constructed so as to include a number like this. I have no problem with that system. I think it's great to invent new math and new number systems meant to rigorously capture a useful intuitive notion, although the construction of the hyperreal numbers requires some questionable usage of the axiom of choice. But I do think it's important to first learn about limits, and how mathematicians made sense out of calculus using the standard real number line without resorting to infinitesimals. It's not a matter of clinging to old systems, it's because limits help to gain a deeper appreciation for structure and character of the real numbers themselves, which in turn will help to understand any extension of those numbers.

If anyone cares as to why the variable "h" is used in the definition of a derivative via the limit as h approaches zero: It's the same reason as to why "h" appears in quantum mechanics. A German mathematian used "h" to stand for "Hilfsvariable" (auxiliary variable) and everyone just ran with it. For clarity, these two occurrences of the letter are not in any way linked. They just happened for the same reason.

Literally Everyone: You can't teach calculus to grade-schoolers... 3Blue1Brown: Watch me.

I did not know there was this much good in the world.

In my experience as a tutor, I've seen that econ students often have a better concept of derivative than the physics students. It's easier to imagine marginal cost (what happens to my cost function when I make ONE EXTRA widget) than to imagine, say, how much extra distance my car travels after a tiny change in time. This fact surprised me.

God, this is my favourite series right now. First year engineering student and it's refreshing to hear everything from a more intuitive angle. I wish schools taught as well as you did Grant.

This is the first explanation of L'Hopital's that has made any sort of intuitive sense, even though I've proven it formally in Real Analysis.

sad this series is FINITE :((((

How should the future generations pronounce your name ? L'Hopital : yes

5:36 "imagine you have 0 cookies and you're dividing them among 0 friends. See? It makes no sense. Now cookie monster is sad that there's no cookies and you are sad because you have no friends"

Thanks to your videos I achieved a 100% in my AP Calculus course!

I have read The Feynman Lectures on Physics. Waiting for The Sanderson Lectures on Mathematics. Grant Sanderson teaches really way, and what I LOVE about him is that he teaches visually, for the sake of learning and understanding, not just for the sake of covering a topic. Thanks a lot. Ur long-time fan, Grant Sanderson.

I absolutely love this channel. it is very selfless of you to create such great content for learners across the globe. The animation, examples and script all reflect the amount of effort you put in to truly inject your passion and expertise in the video. Keep up with the great work!

Content like this makes me confident about the fact that the good part of the internet far outweighs the bad part.

This is the video that has the most utility and is the most accessible. My 12 year old son watched it, understood it, and was profoundly more interested in math after seeing it. And he was already a mathy kid. Well done sir.

Finally! My problem with your previous videos were, that they convey the right intuition for the general case, but leave out the dangerous edge cases! Only in university, through infinite series and limits I began to apply these rules with confidence. It's great to be sure you don't break anything, and I think it is one of the most crucial things to ponder about. What can be safely ignored (and what cannot) for arbitrary choices of some value within a given (changing) range.

Turn on the subtitle at 5:36. A little joke from our great teacher.

I love how every once in a while, rewatching this series gives me different insights on calculus.

This one 19 mins video was worth of the 1 week of 40 mins classes that was given to me in my high school. This does put a smile on my face.

20+ years after trying to understand the teacher at high school I now finally understand the basics of how calculus works, which they never bothered explaining when I was younger. Thank you so much.

Excellent video, as always. Limits are *the* essence of Calculus. The single most important concept to learn in a Calculus course. The only thing I'm "complaining" is: why didn't you actually write down the eps-del definition of limit? You had the ground work all laid down, you just needed to finish it off by giving the actual definition!!! :)

Man, you have honestly become my favorite youtube channel. You have a way of explaining things that give me a "penny drop moment" almost every video.

An awesome video from an incredibly helpful and enlightening series! Watching your videos really fuels my passion to learn and understand. Your clear and thoughtful explanations, along with the fantastic animations, do such a great job of building a deeper understanding and help to expose the beauty of mathematics. Thank you!

I've trying to fully understand all this concepts for several years (+5), you know the feeling of joy that is close to tears? That's how I felt at 17:23. Thanks for the videos, 引き続き頑張れ！

Please make a video on Laplace transform and Fourier transform...

I would buy a pi creature plushie!

Watched this video a few years ago back in like 8th grade. Now that I'm in university and understand the epsilon-delta definition of limits looking back you explain everything really well. Can't find such high quality easy to understand maths content anywhere else on the internet.

Loved these series bro. As a phys graduate, it's refreshing to see someone explain such a "basic" and very formulaic concept in calculus and really define both physically and theoretically what a derivative actually means!

Thank you for teaching me calculus in a way that makes sense to a 9th grader. This makes sooo much more sense than what parents tell me, so thank you!!!

Every educator should show or suggest students to see these videos: You are awesome. Thank you.

Yes please on the probability series! When trying to learn it I encountered plenty of unintuitive (at first) concepts that are just begging for your clear method of explanation. A video on the normal distribution would be great to hear from you :-) thanks for your hard work!

Awesome videos... I am falling in love with Maths until now I was just paying attention on solving problems but your channel videos are making me to see maths out of the box.. Where Maths can be applied in real world . Thanks for the videos.

16:15 3b1B: "Actually discovered by Johann Bernoulli, but L'Hôpital paid him for..." Me: «TIRE SCREECH!!!»

You should really know better than to use \epsilon. Of course, \varepsilon is what all _civilised_ people use!

Comment #3: Thank you very, very much. I have not only understood what you were teaching - finally - but understood a fundamental problem in my earlier studies. You have moved me forward. I am deeply indebted. David Lixenberg

9th grader, studying calculus, studying quantum mechanics and relativity and working on a theory of my own all thanks to this guy. I will owe my knowledge to him. Me and my partner are working on various projects. I love your vids and I support you. Thank you for educating me

Thank you very much Sir, you are an amazing lecturer , I wish I met a teacher like you when I was in high school/university, I wish you will never stop on producing more incredible videos. I recommended your videos to all my friends at the university. please carry on. btw, if it happens that you visit Japan one day, please let me know, I would like to invite you to see the city I live in "Nagoya"

I thought the goals thing at the beginning of the video was a really good addition to the format of the video

I ve been watching your videos since very a long time, but never saw the series in order and completely. I ve never been taught this much intuition for math neither in high school or at the university (and im studying bloody physics). I hope there are more series to come, this is really the best math content there is to find in youtube.

Love this series. With regard to , I usually try to tell my students to think of d_x_ as finite in size, and so small that (d_x_)^n can be safely thought of as zero for n > 1. I explain that it's a notation that abstracts away the limits by writing 'd's everywhere, with the expectation that ratios of 'd' quantities converge in the limit where higher order terms vanish. I'm computationally biased, so I usually follow that up showing ratios of finite differences on a discrete grid, and extend that intuition to the real (or rational) numbers by visualising the convergent behaviour when between any two mesh points, there is another mesh point. But the visualisation of sampling on a mesh helps give the the students the intuition for why behaviour is linear in these limits. NB: I usually teach 3rd year or higher undergraduate classes, where I am repairing poorly constructed intuitions the students may have.

i don't understand 80% of the videos but I can't stop watching them

Thanks, I highly recommend this video to my students. Fantastic!!!!!!!!!!!!!!!1

Thank you very much for your French subtitles . I am learning French and these subtitles are a Big help . Also your wonderful graphics increase my understanding of complicated Math .

The production value and clarity of your explanations makes the abstract concepts in real analysis clear. :D Thanks for your videos

never clicked on a video so fast.

parfait....toujours respectueux !!

4:44 That was a very beautiful detail with the opening and closing tag. It made me smile. Thanks for the great video!!

wow, these animations and your vivid explaing really help me to not only get my brain to grasp that stuff but see the actual beauty in math. Thank you Sir.

I just hope one day you'll do essence of real analysis. Doing this course in my first-year undergrad, and I'm torn apart before even getting to derivatives.

Super video ! Thank you !

These classes are wonderful. Unhappily at the university I had lousy classes about these matters. Now I am starting to understand these matterd. Congratulations, dear Professor.

You are totally rocking it with these newer videos. It's not just nostalgia either. I love your earlier videos. They are amazing. Seeing actual geometry explained by calculus using animations is a true stroke of genius. I imagine these new calculus videos were inspired by your deep learning research. I would love to buy your merch, but I haven't had a job for almost a year and a half. I have 150K (and growing) in student loans and just completely drained my 401k of 60k just to make payments and pay rent. I have no idea if posting this on KZfaq poses any risk. However, if you are looking for inspiration in your future videos. I would appreciate any financial mathematics animations you would dream up. I am not the same person who took out so much money. I however still a person. Thus, I need all financial advice I can consume. I don't think there are haves and have nots, I think each person's finances are either accelerating up or down. The rich get richer because they are accelerating up, the poor get poorer because they are not accelerating up consistently. Right now, I am still accelerating down. Compounding interest is a lot like gravity. One last thing. If you were to self publish a book using some of these animation frames. You might usurp all other math books because you would have a link to the videos the pictures are from. Thus, giving the reader the option to test their knowledge on the text and if that is not enough intuition then they would have the convenience of watching these artful videos.

I've somehow gone my whole life without having an intuitive understanding of L'Hopital's Rule. Now it makes perfect sense. Thanks!

Cool. I never understood why L'Hopital's rule works. Not anymore!

I love going back and re-watching ALL of your videos as refreshers, but I especially love this series.

My comment on Chapter 6 has just been shown now for the first time at 2:27, the Greek delta. Great graphics and deep explanations in these videos and I love it!

7:47 That made me laugh.

6:22 that 'come on' is so relatable😂❤

I know calculus and analysis are similar to each other, but it'd be so great to see a series about real analysis like this as well! You are the best!

I just wanted to say thank you for all your teachings! You are a truly gifted teacher!

I don't know why, but I have learned about derivatives and limits a bit different in my country. We started with limits to be able to define derivatives and we were solving them differently, which really showed the beauty of it. We mentioned l'Hopital's rule briefly, but I had to see for myself how useful it is in harder limits. Also we didn't write the fractions df/dx at the end of everything, the calculations were much more clear thanks to that. However, we started using it at the end of integral, to know what is the variable. I guess we would have used in normally if we were planning to get to multivariable function derivatives. Also, implicit differentiation was a bit different too. This series is great, but it really shakes with my view on calculus, that I have built for the last two years.

Add some more t-shirts with Pseudo-Hilbert curves, or the contour lines of the Riemann-Zeta function to your store.

This is one his only videos where I’ve actually had a proper intuitive understanding... finally

Thanks a lot for making such great animations. I am truly grateful for your work. I probably would have spent an hour trying to understand the epsilon-delta definition if it was not for your video. (PS: I am a fresher in college) THANKS AGAIN

this is like watching art. love it

If I keep pushing my upper and lower body to the limit, I might end up in L’Hospital... :D

Love it, these courses saved my life a lot, I keep watching those!

I am very impressed with your videos. Looks like there is no limit in what you can teach us. Great job.

PI CREATURE PLUSHIE Thank you for the intuitive description of L'Hopital's rule. As a university student I think that it was the most enlightening part of this video.

Any plans to create an essense of stats / probability playlist?

Man thank you for all your videos, you make maths easy talking softly and explaining graphically.

I can't believe you explained epsilon delta definitions in 5 minutes, while my professor couldn't in a year... thank you

4:45 perhaps an analogy? Imagine you have a friend whom you have no idea where she lives onand you want to meet her, though you're not allowed. Met at college. So you ask a friend, and she points out it's in the next city over, so you head there. Then arriving at the city, you ask another friend, she says it's in a particular neighborhood, so you go there Finally you ask again and you're told on what street she's in. Now you can't go there yourself and go into her house, since her father so grumpy, but at least you can see from a distance where the house is. Limit is seeing the house in a distance and saying I know enough And infinitely small is when you're welcomed to the house. Both scenario leads to knowledge of the location but You don't need to enter the house to know where it is, so you wouldn't need GPS to tell you next time. *I know it sounds creepy but it's a weird analogy I come up with that might help? Like limit is the direction of which everyone is pointing at The GPS is manually checking actual values nearing up to the location

I'm using this to prepare for my upcoming bachelor degree in maths! I am done with my Abitur (somewhat like a HS degree) and start going to university in a few weeks

Thank you my guy 😭 Noone explains this properly and i was just going from here to there trying to find someone who'll properly explain and then your video popped up❤

These videos are worth watching and very informative. Thanks for clarity in concepts

Great video! I'm curious -- what are your thoughts on nonstandard analysis, which made "infinitesimals" rigorous? I thought your earlier videos in this series were very infinitesimal-friendly, so I'm kind of surprised that you're taking a stance against infinitesimals here.

Who always read that as "hospital" like me or am I the only one?? Anyways, thanks for this brilliant video.

Hehe this brings me back to my first semester of calculus, our prof. had us doing delta epsilon proofs! I asked my cousin with a mathematics degree for help, and she sent me an image of the definition from one of her old books. People would call him mean for having us do those where all the other classes would skip over that part in the text, but I really value that part of my journy into calc I don't think I would have done as well in the following semesters of calc.

I LOVE YOUR VIDEOS SO MUCH THANK YOU!!! I was struggling to understand L'Hopital's rule and you cleared everything up for me. Thank you so much!!

Great series and animations. One thing: Sure there is "something new on the conceptual level"! What you show is not a formal definition of the limit "L := lim{h->0} e(f, h)" in the way of "x := 2+3" (or "x:=SS0+SSS0" in Peano arithmetic) or "f(x) := x^2". You can't easily tell beforehand if a limit exists and thus provide a domain. The logical sentence "lim{h->0} e(f, h) = L ⇔ P(L, f)" is as close as you get to a "definition of the limit" and it implicitly captures a number or it doesn't. Everything else (e.g. writing, for general f, df/dx followed by an equal sign) is abuse of notation that you don't find in a logic text where all statements are closed (by universal quantifiers). By the was h is often the "Hilfsvariable" (helper variable) in German text. The Planck constant was termed the same by Planck. Keep up the great work.

i'll sag my teacher to show these videos whenever we will start studying calculus (4 years :P), books just give definitions over definitions, but this is very intuitive

If we were stuck using the limit process for derivatives during Calculus, I don't even want to know what it would be like having to do that! Great videos as always!

An invaluable resource for both tutoring freshman calculus and meditating on basic concepts in preparation for graduate level analysis.

Dropped out of Calculus my senior year in HS because I didn't know what was happening. Your videos make it so much more intuitive.

I really think this should be before derivates, it would make explaining them a little easier.

I m much lucky that utube has u type of explainer...I wasn't able to understand calculus in mine's institute...but urs vids r the best to know about calculus THANK U SO MUCH

I don't have enough words to say how grateful i am for this video.....god bless you seriously

Sometimes you just look at these videos and feel the need to give a clap...

please include your logo and a math puzzle/oddity on the shirt. The logo looks like a mesmerizing eye, very visually interesting

I am loving this series! Makes me excited to be taking PDE next semester

I just gotta pause the video halfway just to type this so I can tell how great your videos arrreeee. Bless youuu brotherrr

That bernoulli dude got some wild, wild hair

9:49 I really don't know why that was so funny

That was a nice simple explanation for L'hopital's rule. I've never spent time examining a proof of the theorem, but I have wondered about it, and you showed that the limit of the ratio of two functions, in the cases where both functions go to 0, is essentially the ratio of their values close by (at least when this limit exists), and this is exactly what the ratio of the derivatives converges to also: lim as x2 ---> x1 [f(x2)-f(x1))/(x2-x1)] / [g(x2)-g(x1))/(x2-x1)] = lim as x2 ---> x1 (f(x2)-0) / (g(x2)-0) = lim as x2 ---> x1 f(x2) / g(x2) = f(x1) / g(x1).

At first, I didn't understand anything. However, after reading about the same topics in book 'Mathematics for ML' and 'Dive into Deep Learning,' the puzzles started to make sense. Thanks for such an amazing video.

I love how mathematicians delved into ways to find the infinitely small - calculus truly is a beautiful subject.

Nice video but I don't like the bashing of infinitesimals. Infinitesimals are rigorously constructed and proved by Robinson's nonstandard analysis There are no logical fallacies with infinitesimals. And given the history of calculus, perhaps more intuitive. Also within constructive mathematics (specifically smooth infinitesimal analysis), nilpotent infinitesimals are valid. Computationally, these infinitesimals have been reified as dual numbers or nilpotent matrices. Limits do make sense in numerical computational methods

Ooooh I'm so close to the integral video. Thank you for the series!!!

I watched this series in spring 2020 during lockdown when I didn’t even know how to ‘differentiate x’ because I had never touched calculus. Now over 2 years later I am going to university next month and find myself coming back to these topics I never learned in school.