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

[@devinneal438]

I'm genuinely jealous of the next generation of students who will have these videos to introduce them to calculus. The understanding and intuition they bring to a newcomer must be really exciting.

[@eduardocortez476]

Devin Neal It is!

[@JannikPitt]

I was first exposed to Linear Algebra by his video series. It really got me hooked and I looked forward to every video curious about how the subject will evolve. That was really awesome! I bought some books on mathematics and self-studied maths quite a lot since then.

[@adamcummings20]

I'm going into a-level further mathematics next year after GCSE and I am so grateful that this guy does what he does so well

[@omamba5105]

They're exciting for me as well, and I really enjoyed calculus back in high school.

[@lizardbaron3727]

Devin Neal they're very exciting

[@Treegrower]

Grant, I don't think I've ever watched a 20 minute math video and wished there was more. You are super talented, and I want to thank you for releasing this to the world for free. This is not just math. This is art.

[@petercoool]

ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ He is Dr. Art

[@francescosorce5189]

Math is art, and he's an artist

[@joaquinmela5781]

His videos are just so awesome that I'm gonna cry

[@aniketh_99]

I want to like this but you have a perfect number of likes and I’d rather not break it

[@hziebicki]

Math is art. And art is math.

[@oliviacranmer-gordon3112]

When I watched this years and years ago I fell in love with maths again, now I'm a maths teacher and I'm passing this down to my students. I'm so grateful for your channel and I hope you understand these are more than just KZbin videos, they mean a lot to people.

[@3blue1brown]

That's incredibly touching to hear, thanks for sharing!

[@divy-07]

​​@@3blue1brown respect for replying after 6 years since the post date

[@theseangle]

Just imagine being 3Blue1Brown and reading such comments from time to time. Just imagine.

[@shivendrasingh6136]

​@@3blue1brown Grant, It was just wow ❤❤... Since you've got this exceptional talent of explaining and animating So why not try some topics like The Essence of complex no....

[@AbdulKalamabdulkalam]

That's it, I'm promoting this channel wherever I find people who say " I'm really not a math person"

[@uzairakram899]

I had the same idea but some people are really not into it.

[@joshtipton7417]

I was 'not a math person' in high school because I had a crappy understanding of algebra and equations, but always excelled at the theory stuff regardless. I eat videos like this up because you learn all the theory and abstractions that show how cool math can be, but someone else crunches the numbers for you. They should really use this as learning material in schools, it could help a lot of students like me.

[@Zivlix]

Josh Tipton as a person like you who loves theory and abstraction but not so good at number crunching but is about to be put into the meat grinder of high school math I cannot say how much I am glad to be at this point of history, a point at which pure math and implied math begin to cross along with so many people and resources to help me to grow my understanding. If you have not then I recommend this video - kzbin.info/www/bejne/hZ6tXmd4YtKGY7s

[@dimdenEFF]

I'm really not a math person and I don't understand it ;_;

[@tejpatel5320]

I think you will be surprised at how much someone would still dig in their heels. Learned helplessness is a very deep hole made be habit of thought and preconceptions of their ability and of the subject matter.

[@ramitello3156]

"I would use the letter "d" for distance, but that guy has another full time job in Calculus" Lmao

[@gregorybattis9588]

I was thinking I was the only one who would laugh at that.

[@rka8215]

Isn`t "d" a minor? Isn`t it illegal to make children work full time?

[@rka8215]

@@teeraxgaming Isn`t "d" "D`s" son?

[@rka8215]

@@teeraxgaming Save the children!

[@rka8215]

@@teeraxgaming Why??? Are you kidding? Because 1. They give young parents extra time at night. 2. They make middle aged parents go back to calculus! 3. They keep mature parents from buying a Porsche or a villa in Italy and invest their money in college tuition. 4. They help old parents formulating their last will. 5. And they keep pediatricians happy. That`s why!

[@paulmatsumoto618]

Thank you for the great quality videos. It makes things easy to understand.

[@mainconman2201]

Fr

[@nikthedrought]

ong

[@sandhan1t1zer]

for a second i thought that it said $1000

[@johnchen2325]

@@sandhan1t1zer But it's worth more than a thousand.

[@abhilashasinha5186]

@@johnchen2325 sand hanitizer?

[@erndiniz]

Grant, I'm a 39 yo man in the verge of changing my carrier from the Ad industry to the Dev and AI realm. I really wanna thank you for your overwhelmingly great work. This video, oddly enough, changed my life, and I'll forever remember you as a positive force for this change. Thank you very much.

[@木心-x3j]

Ad AI.wow

[@GeodesicBruh]

You could almost say that after this video the derivative of your feelings with respect to time is positive

[@ritageraghty4404]

I am 53 and I am just relearning maths as a hobby. I'm doing it for 3D art and animation including physics simulations that require maths.

[@rahulupadhyay3570]

All the best

[@krisjaniskalans3474]

man this is fucking powerful

[@firen777]

I'm not sure which one I find more amazing: the math itself, how clearly you explain it, or work you put into the animation.

[@sakuhoa]

Both

[@Anonimowany1]

WIsh I had all these KZbin Videos when I was still at school like 10-5 years ago. I loved math and physics. (I studied math and physics at the university afterwards) The way those subjects were taught at school was plain and super boring. All these videos on KZbin are so fun and I learn even now from all these. I believe in the next decade or so - some really clever minds will arise - smarter than Einstein and Stephen Hawking and all others. Just due to the fact the way all the knowledge is provided and super easy to learn and visualize with the current age of technology. It literally wasnt like that even just a few years ago. The future is looking bright.

[@deedewald1707]

It's all amazingly awesome !

[@razvanefros411]

0: Oh? You're approaching me? dt: I can't beat the shit out of you without getting closer

[@sadkritx6200]

0 : Oh ho, then come as close as you like...

[@RummanNaser]

I love this community. Making jojokes on maths xD

[@rohankishibe8259]

JoJo references on maths related topics is something i expected when certain zeppeli started using the golden ration on us mortals, then mr Jonathan started flexing his spin.

[@bluebrowney251]

This was hilarious

[@felixgutierrez993]

Oh shid I never thought my JoJo fascination was gonna follow me here in Calculus 🤣

[@tannerkent5446]

Those "ohhhhhhh, aha!" *click* moments that were so rare in my University calculus classes are so frequent in these videos.

[@methatis3013]

University? I thought you learn this in 3rd grade of highschool. Im in 8th grade rn. Do I srsly have to wait 5 more years for this?

[@primalforlorn]

@@methatis3013 Not everyone has American style of education system, Calculus is elective in my country for example

[@RajShekhar-jy2zi]

@@methatis3013 i study this in 11th class in india .

[@Georgexb]

PrimalForlorn In the UK, calculus is introduced in sixth form (year 12), but of course it carries on to university level

[@Anonimowany1]

WIsh I had all these KZbin Videos when I was still at school like 10-5 years ago. I loved math and physics. (I studied math and physics at the university afterwards) The way those subjects were taught at school was plain and super boring. All these videos on KZbin are so fun and I learn even now from all these. I believe in the next decade or so - some really clever minds will arise - smarter than Einstein and Stephen Hawking and all others. Just due to the fact the way all the knowledge is provided and super easy to learn and visualize with the current age of technology. It literally wasnt like that even just a few years ago. The future is looking bright.

[@minutebrainperson8324]

3Blue1Brown, I'm a master's degree student in theoretical physics with 1 year left, and probably not in your intended target group. Despite many of your videos covering "basic" (read: essential) topics, I find them fantastic tools for refreshing my knowledge, and even learning something new. For example, from your series on linear algebra, I learnt to visualise the columns of matrices as transformations of the respective unit basis vectors, which has been a great tool for when I have been reading group theory and quantum field theory, even though it was such a simple observation. Thank you for these videos! I look forward to more!

[@ElchiKing]

Well, I would consider myself a quite good math student and yet I can still learn much from these videos despite "knowing" most of the content already.

[@Krunschy]

+Elchi King Yeah, even though you already know through theory what the formula of the derivative and such things is, having such a great visualisation of it, really makes things easier when trying to learn and picture new you learn related to that topic.

[@kylepoe5139]

Imagine that you yourself are a graduate student in theoretical physics. You want to express your appreciation for a video, but you want to actually frame it in the context of why you appreciate it. Just saying "cool video dude" doesn't really do much. Taking the time to explain why something is useful despite not being a member of the intended target audience is far more useful to the man who makes these videos, and it is undoubtedly nice for 3blue1brown to know that his work is appreciated by people in academia as well as high school students cramming for their AP test. If someone pursuing a degree in theoretical physics cannot even state what they are doing with their life without apparently sounding like they are bragging, then it is truly a sad state our society is in.

[@mikehaskel8455]

I agree with Kyle Poe. There are plenty of people out there who, beyond being good math students, live and breathe this stuff every day as part of their profession, and really do pretty much know everything in these videos already at a deep level. After all, the author can't be the only one in the world qualified to make a video like this, can he? But these videos are still so well constructed and explained that they're worth watching for a fresh take.

[@joshuapollard184]

In one of the linear algebra videos when he said eigenvectors stay on their same span during a change of basis which means that they're just the axis of rotation, I think my jaw literally dropped

[@phsaraiva19]

I'm a mechanical engineer, 37yo and have a carrier lasting over 15 years. This is by far the best math class I ever had in my entire life. Thank you very much, your are not a math professor, you are an artist.

[@parispapadopoulos8892]

"The Essence of Combinatorics" is something I would really really love to see from you! Keep up the good work!

[@pratikshayadav9279]

Yes please please

[@aniruddhasanyal7625]

Yup

[@useltrummis]

notice the point and draw a line.... time is created from 0 to 1 too... i dont remember a nicer conciser version of "how to press start" dug deep, dude

[@aarushiaiyyar]

Yes yes yes please. Please cover vombinatorics and probability.

[@High_Priest_Jonko]

Fact: Most adults could never pass a course on Intermediate Combinatorics

[@gracemalcom7358]

I have a bachelors degree in mathematics, and even though I was a straight A student, I never really "got" why we did what we did to solve problems, just that we had to do it.. until your videos. Thank you for giving me such clarity on a subject I love with all my heart and a full understanding of why problems are solved the way they are! You rock.

[@cubicinfinity]

I started watching these videos when I started learning calculus but I stopped because it conflicted with the way I was being taught. It's a better experience after having gone deeper into calculus and watching these again. I'm solid enough in it that I can keep track of the different mindsets.

[@sidharths9416]

@@cubicinfinity You are absolutely right bro. Me too had the same experience. In fact watching these after all that taught in our schools/colleges is really making us fall in love and admire these concepts

[@nerd2544]

@@carlgauss1702 How did you get a job then?

[@leelaanandabhavan3083]

Well....I think you suck.

[@the.abhiram.r]

@@sidharths9416 my teacher just gives us a bunch of formulas lol

[@dripcat5940]

everyone: "no one can calculate instantaneous change. it's totally nonsensical" fathers of calculus: "so I'm gonna do what's called a Pro Gamer Move-"

[@deedewald1707]

The art of problem solving is solved here, no problem to show !

[@RF-fi2pt]

Wrong. Makes sense so well as work at derivative. See My 3 years ago post, also i posted at keystone video about paradox of derivative.

[@Brahvim]

*Pro Mathermatician move

[@user-yj8uv7gi3o]

@@Brahvim *Pro Mathematician Move

[@Brahvim]

This comment is now 2 months old, thanks for the correction, @@user-yj8uv7gi3o 😂👍 (Saw this after 3 hours :|)

[@Carsonlego]

It literally took me months to understand this video but I’m super glad I do now. Thank you so much 3blue1brown

[@ozkaa]

watching this series has made me fundamentally question the quality of the education I received at school growing up. It's exposed how badly we were explained these concepts in class. I did so much calculus at school without ever being given a real understanding of what i was doing. And then I wonder if the same maybe applies to all my subjects at school. maybe my whole education was actually entirely sub par and I've fallen far short of my potential as a result lol.

[@robinlarsson2108]

Welcome to humanity!

[@imperialeagle564]

@Mr Right yeah thats very right mr right

[@tolep]

@Mr Right Also, watching this, people are relaxed, can pause and replay etc.

[@DMahalko]

@Mr Right It should be possible for 12 year-olds who think they have all the answers to go out and get a job, an apartment, etc. If they want to continue their education later as an adult, it is still available and fully paid like grade school was.

[@wallacegrommet9343]

Your teachers were mediocre and apathetic

[@keztaylor4306]

I'm studying engineering 10+ years after leaving school. This series is a blessing! I never took calculus or advanced maths fomally and I could not properly grasp the concepts through the book material. Seeing this laid out in such a clear, visual and logical way is sublime. Thank you for taking the time to create such in depth videos.

[@bamunua]

I definitely agree with you

[@anandkrishnan7997]

Being an undergrad student, studying at a level a little more advanced than this, your videos really make me take a step back and appreciate the beauty, a sort of 'stop and smell the roses' for math. Thank you, 3Blue1Brown for being a brilliant educator.

[@xanderlewis]

Same here. I first watched this video when it was published almost exactly two years ago, and coming back to it now is something really beautiful. At the time, whilst I could follow it and gain a lot of intuition, it seemed quite complicated and it was difficult to really see how it fit into the ‘bigger picture’ of mathematics. Now I’ve nearly finished my first year of undergraduate mathematics and been introduced to the more general principles and some of the more rigorous constructions of calculus, this is just wonderful to see again. It’s encouraging to see how much progress I’ve made personally in my understanding but also to see how almost exactly the same techniques used in this series can be applied in so many ways to obtain all sorts of other incredibly beautiful results in other areas. The fact that much of mathematics is so ‘useful in the real world’ seems like a wonderful coincidence, because even if it wasn’t... it’s just so....... lovely. ♡

[@Fyizze034]

I'm a french engineering student, and despite the fact that I already know these things, I love watching your videos because your way of talking about maths and illustrating them really makes me wonder deeper questions than what I already learnt. Great job I love your channel 🙃

[@ChiChiStrawberry]

Evi1M4chine This probably sounded a lot more passive-aggressive in my mind than I think you intended it to? I would say that engineering students do understand things deeply, but there will always be people, in any field, that simply accept rules and move on. But again, that's only a few people, not all people. Videos from this channel are made for the purpose of curbing that, however, and that's lovely! :)

[@zeronothinghere9334]

Paragraphs please. My eyes hurt. E: On the other hand we've got to ask this: Should education teach them how to live, or should it teach them how to think?

[@Fyizze034]

Evi1M4chine I 100% agree with Yue Chi K, I think in any field there will always be people trying to cheat as you said it properly, but they won't be good engineer, surgeon, teacher, etc... According to me, some professions require passion or at least interest to be able to work in correctely, and engineer is one of them, I hope your flatmate won't realise it too late.

[@vampyricon7026]

+

[@BGbaas]

Wow I've almost obtained my bachelors degree in Aerospace Engineering and I finally get to see why the derivative of a polynomial is the way it is, thanks for your great videos and insights!!

[@theguybroseph]

This gives me hope hahaha

[@tuneboyz5634]

@@theguybroseph thats great little Ben

[@fernandotanase114]

Can you give updates on the searched/found jobs once you've got your diploma, please.

[@jebediahkrimsoncraftleding3012]

Embarassing.

[@Tech_Gamers]

Thank god ,we're all safe to fly in your airplanes now💀

[@ToddWCorey1]

In college my Calc 1 professor led us to the point where we could see the derivative for ourselves. When, in a homework assignment, I did just that, he implied that I must have taken calc in high school and been pretending to have had that breakthrough for myself. A real kick in the jibblets. Some people just shouldn't teach. When I read your intent for this video series, it really hit home! Thanks for allowing folks to experience the wonder of discovery.

[@arnouth5260]

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.

[@redpill6379]

At school they teach u for exams that's why.

[@chipou812]

59 years old and Grant is taking me to school. Thank you!!!

[@meereslicht]

Here I am, Professor, taking notes as if my life depended on it...Thank you very much for an excellent, much expected and very illuminating series!

[@jerry1364]

Your work here has permanently changed the way I think and how I feel about Mathematics and life in general. Please keep up the good work. When you feel down, please remember that there are so many of us that are grateful of what you have done here.

[@deepakg8758]

omg this channel makes me think how poorly these concepts were explained in school. Keep up the good work.

[@Mr1987atul]

The people with such deep knowledge will ask heavy salaries. Bcoz they know their worth. And schools cant afford such teachers. Which is why we get poor education from teachers with poor knowledge.

[@lokmost]

Nice video. My University's teacher used this to intoduce us to derivatives

[@Anonimowany1]

WIsh I had all these KZbin Videos when I was still at school like 10-5 years ago. I loved math and physics. (I studied math and physics at the university afterwards) The way those subjects were taught at school was plain and super boring. All these videos on KZbin are so fun and I learn even now from all these. I believe in the next decade or so - some really clever minds will arise - smarter than Einstein and Stephen Hawking and all others. Just due to the fact the way all the knowledge is provided and super easy to learn and visualize with the current age of technology. It literally wasnt like that even just a few years ago. The future is looking bright.

[@rohan1002]

If I only had the same teacher.

[@primeroyal7434]

He/she should get a reward.

[@primeroyal7434]

He/she should get a reward.

[@cantseeme9886]

uhh, me tooo ahaha

[@animeonperseest]

I think it's safe to say that by impacting millions of people watching your videos, you've single handedly made the world a little bit better place.

[@newton21989]

I took calculus in college. The professor explained how to find the derivative of a polynomial. I didn't learn until many years later, after I graduated, what a derivative actually is.

[@No-cg9kj]

Then you just didn't pay attention or read the book.

[@abcrtzyn]

I “learned” derivatives at school today. My teacher said to write down the definition on the board but not the crossed out portion which read “it’s almost like finding the slope at a single point.” And then he gave us this formula, f’(x)=lim(h->0) (f(x+h)-f(x))/h. I spent the example problem time as time to figure out how this equation gets the derivative. Thank you for showing what derivatives are and not a random equation that means nothing to most people.

[@jorriffhdhtrsegg]

At least you remember it. I am sure we had a 'slope of line on curve' lesson and think "um ok why, just playing around with tangents now?". Well its only the foundation of classical physics, involves infinite limits and all sorts, but anyway...if only i knew at the time instead of "No JuSt LeArN iT"

[@dank6617]

This reminds me my high school teacher who was really great at teaching, instead of force feeding us formulas to remember, she would first show the proof of why it is what it is. The first time I saw a mathematical proof it was mind blowing. Unfortunately not many math teachers actually even know proofs, and most kids around the world are getting gavaged with rote memorization of terminologies and formulas they don't understand... :(

[@evanchong6482]

tbh not everyone needs the proof, alot of us just studied maths to pass and we really couldnt be bothered to learn the history and context

[@RenatoKestener]

@@evanchong6482 that is exactly the point. "Studying to pass" makes no sense. It is the same as saying "living to die". Study should be about proof, about curiosity, about exploration. This is what makes humans a miracle and not slaves of some nonsense way of living...

[@mistec34]

I love that even though I've been an avid student of math for most of my life, preaching its wonderful usefulness to anyone who will listen, your videos will invariably contain 50% of things I have learned, and the other 50% wisdom that is gained only by tying all fields of math together to produce a bird's eye view of a master concept. That extra amount on top of the base material represents you trying as eloquently as is humanly possible to share your bird's eye view with the world. We are fortunate to have your passion and knowledge used in this way, and we can only say "thank you!"

[@3blue1brown]

Next up with be "Derivative formulas through geometry". Full playlist at kzbin.info/aero/PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr Some commenters have pointed out that the way many actual car's speedometers work is to induce a current with a magnet whose rotation is determined by the rotation of the wheels. Given that the laws of electromagnetism are written in the language of derivatives, this is a nice example of measuring the derivative directly, not as an approximation. Neat! The broader point in the video, though, is how trying to make sense of "instantaneous change" is intimately tied to the function's value around that point. That is, the information that goes into a derivative at a given input is not actually confined to that input alone.

[@amo6139]

great video

[@raymondsutrisno9032]

There is a mistake in the notation at 13:30, shouldnt it be ds/dt of s(t) not ds/dt t. or d/dt (s) or d/dt of t^3

[@Quacky_Batak]

At, 9:13, how come you calculated the derivative at t=10, isnt the graph discontinous?, also, by your intuition(which is best), we can't extend ds "just" above t=10. BTW great video love ur stuffs!!

[@zairaner1489]

The graph isn't discontinous at t=10, we just looked the function in the intervall (0,10) and it isn't even clear wether 10 is in the intervall or not. You can define the derivative even if the intervall is closed, in which case you just have the limit from one side in the boundary points

[@zairaner1489]

dt is not a number! Thus it makes no sense to say its zero, and also no sense to subtract soemthing from it. It's just notation for the limit

[@jeffgalef121]

Your animation bringing the secant line down to a tangent line really drove it home. Thank you so much.

[@oliverlawson3970]

I'm 14 and love these videos-they really help me understand calculus though especially since this is a series I would say a bit more pausing and giving the viewer a chance to work it out themselves would be helpful to keep engagement, but apart from that these videos are amazing and I love this series, even six years after you released it! Thank you so much!

[@PriggarGaming]

Great now even a 15 year old understand calculus and i am here i suppose a bit too late lol Seriously tho Are you now actually able to understand so that I can have some actual review even tho there are 100s of comments saying it is good...

[@tahmidt]

The amount of times I went like 'oh' and 'aha' and 'wow' says clearly about how much of an enlightening experience this was. Thank you so much!

[@suvrat]

The choice of colours for “instantaneous” and “change” was really clever. I love your attention to detail. Thank you so much!

[@feishao8892]

Thank you. I am 52 years old. My son recommended your videos and believed I can learn Calculus with you at the my 50's. And he is right!

[@ArnauViaMartinezSeara]

Being a mathematician working on the financial industry (and despite what you might have heard about financial markets using maths quite often and rigorously) who wants to recover part of my former mathematical reasoning and intuitions lost among nonsense, nonrigorous and nonscientific economic pseudo reasoning, I couldn't have stumbled upon a better resource than this channel. I will only say congratulations, because no matter which words I pick, they will do no justice to the art performed here. As announced, congrats!

[@atomisedman6235]

Thank you so much for calling out the "instantaneous rate of change" oxymoron. I pointed it out in to someone and they overruled my point simply because they pass exams better than me

[@maymldrz]

This is great! I used to study maths at school in such a bad way, through memorization and solving problems in an "automatic" way, without studying the actual reason and meaning of mathematics. Now I'm at university, reviewing again some fundamental topics and I'm seeing maths in a completely different way. Your videos are great and are helping me a lot to discover new things. Keep up the great work! Ps: Sorry for some grammar mistakes, I'm learning English.

[@iosefka7774]

my mind was literally blown when he explained how you can just... ignore terms containing dt so intuitive!

[@lukaszspychaj9210]

I had to stop the video at that moment. Wow

[@Brahvim]

Same! (I realized how you can always break down a fraction later)

[@DoesMahBlockLookBig]

As a precal student, your videos actually hype me for calculus next year. I love to be ahead and have a clear understanding of calculus going into school next year.

[@terner1234]

@@DoesMahBlockLookBig what didi you learn in school until this point, did you get to integrals?

[@DoesMahBlockLookBig]

star of duty Just yesterday we touched up on the basic definition of a definite and indefinite integral. Right now we’re taking baby steps by using Riemann Sums and Trapezoidal Rule to help us visualize how integration works.

[@terner1234]

@@DoesMahBlockLookBig since english isn't my first language, I don't completely understand what you're talking about. I don't think we even learned what integrals are except for "the opposite of derivatives" or "the area between two graphs"

[@Ashebrethafe]

@@terner1234 Riemann sums are what was discussed in the first video -- an approximation of the area under a graph as a sum of the areas of rectangles, calculated by (a) splitting the part of the graph for which you're measuring the area into several pieces; (b) choosing a point on each piece of the graph; and (c) calculating what the area would be if each piece were replaced by a horizontal line passing through the chosen point. (If I recall correctly, the actual definition of an integral is the value that the Riemann sum approaches as the width of the widest piece approaches zero.) The trapezoidal rule is a similar idea, but with trapezoids instead of rectangles -- instead of replacing each piece with a horizontal line through a single point on that piece, you replace it with a straight line connecting the endpoints of the piece.

[@alexovichsky]

Then you will be that annoying know-it-all student who sits in the front and tries to lecture the professor.

[@RishiSundararajan]

He makes magic using just a bunch of living pi symbols with eyes and some really good graphical animation. Hats off. Never going to forget how you helped my in my quest of ultimate knowledge. Thank you. Makes me question what schools even do.

[@sir_albaxious1909]

Relatable with me. I wanted to learn calculus because it just passed through my mind one day. I thought I would learn this after a some years and that learning it is impossible. But as I begin learning, I realised that this branch is the most beautiful branch in all of mathematics. And yeah, I learned this for the purpose to use Integrals in my work and for ultimate knowledge.

[@emmawood9917]

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

[@uzairakram899]

ese est percepì

[@AthAthanasius]

"If you can't clearly, and succinctly, excplain a concept to someone else, then you don't truly understand it yourself." - Paraphrasing 'someone'.

[@montanasnack7483]

@@AthAthanasius Thats exactly what i wanted to write

[@daywill8849]

@@AthAthanasius Richard Feynman

[@fahdal-sebaey3322]

@Emma Wood .. Do you mind if I steal this quote for my book, desktop, discussions.. and pretty much everything?

[@benvarner4011]

The music + Grant’s voice + the visuals + the content make this channel my favorite math KZbin channel.

[@lacieemai6103]

I've almost burst into tears during watching this. The derivatives have never been so clear to me. Thank you so much, exactly what I've been looking for!

[@mauriziop4307]

since is so clear for you maybe you can help me .sorry for my stupid question .in this video the derivate is 12 ..but 12 what meter for second ?it should be the istantaneous speed and so how is possible it s the same at any point?thanks in advance

[@fehgmenshcr2427]

​​@@mauriziop4307The slope of the tangent line to the graph is 12, he just ignored all the terms with dt in them because dt is approaching 0, so it's a REALLY small number, but not 0 and not infinitely small too. So those were safe to ignore

[@erikbeserra32]

10 days of Calculus videos, 10 days till the AP Calculus Exam. Coincidence? I think not!

[@nestorv7627]

lmao, I'm just waiting for him to explain deeper Euler and Newton's method, and the Lagrangian Error Bound.

[@SpaghettiToaster]

newston's method is not an essence of calculus

[@williamshearer5506]

Erik Beserra no

[@S404_44]

yo soy arból 5 lol is the second-worst grade here

[@eldesconocido5992]

Erik Beserra .

[@barryhughes9764]

We are truly honoured and privileged to have access to the thought processes of the giants who have gone before.

[@ThinkTank255]

"Sometimes when everyone is standing on shoulders of giants, it is better to be an ant." - ThinkTank255 What this means is that it is great to make progress based on the works of others, but sometimes, we must challenge and question even the giants to make progress. Remember, these giants were humans, just like you and I. The idea that we cannot improve on the foundations of mathematics itself should never be assumed simply out of honor or veneration of the great men that came before us. Indeed, I think many of them would themselves be honored to be challenged and even proven wrong. I think mathematics has, to some extent, lost its way. It has been standing on the shoulders of giants for so long, nobody knows how to be an ant. There is a lack of critical thinking, even within mathematics itself, which is actually quite frightening. To put it metaphorically, standing on the shoulders of giants leads one to believe that maybe the giants shoulders are the only way to elevate oneself and maybe no mathematics exists where giants don't exist. In contrast, I think mathematics exists everywhere, and we can invent new mathematics and new ways of doing things if we try hard enough. We need not be bound to the dogmatism that exists in mathematics.

[@asemhisham3507]

Well actually... Human brains evolve over time... Meaning most of us are smarter than newton and the others.... But we are a light year behind their determination... Many of us are just too lazy and thats true and its a fact

[@adeelali8417]

@@ThinkTank255 It's not convenient for a person to do this though. You're right though there may be other crazy math that we will never find.

[@christina.sherman]

THE 👏🏻 FACT 👏🏻 THAT 👏🏻 THESE 👏🏻 VIDEOS 👏🏻 HAVE 👏🏻 NO 👏🏻 ADS👏🏻

[@sameendusk2623]

"when you bind yourself to all but just a single instant, then there's not really any room for change." true in math and in politics.

[@MrOdsplut]

Sameen Dusk We live in a society.

[@mannyheffley9551]

Are those any different? hMmmmmMMmm

[@squibble311]

ew politics

[@calebjohn6159]

@@MrOdsplut we live in a society

[@nagarajuchukkala9538]

One thing...Please don't stop making this kind of videos🙏

[@RishiSundararajan]

He has truly achieved his initial goal of the series. I do feel what those mathematicians would have felt just after discovering derivatives and the unique formulas for a few of them. Congratulations. Hats off.

[@alexfernandez4883]

I have no doubt that this is one of the most clearest and satisfying fundamental math topics videos available in the planet. What an amazing useful and enriched piece of content. Thank you so much Grant, now I have totally understood this concept on a intuitive way

[@gregodify]

When I learned calculus, for t>>0 instead of d(t) we used Δ(t). This notation was helpful for me to understand the concepts you discuss so well.

[@miholju]

"Sleeve of Wizzard" - Borat

[@vincentstone7272]

My calculus teacher teaches us that the reason we use "d" intstead of (delta) is because d is latin script and delta is greek script... don't know if thats helpful

[@19divide53]

@@vincentstone7272 Except in mathematics, except for the conventional ones (e, π, special functions, etc.) you could use any of the Greek scripts or Latin scripts as symbols for mathematical objects. It's perfectly valid to write τ^2-τ-1=0 or "let Λ be a connected open set" or "Θ(x,y)=χ(x)ζ(y)".

[@quinntolchin3080]

@@vincentstone7272 @nomi udo Δ(t), represents "a defined change in t", whereas d(t) represents "a change in t that approaches 0". Grant used d(t) as if it was Δ(t) in this video for informational purposes, but really saying that d(t) ---> 0, is redundant because that is what the "d" represents. Whereas the "Δ" just represents a change that does not necessarily go to 0 unless you take the limit of it. In other words d(t) is a shorthand notation for the limit as Δ(t) approaches 0.

[@hunainahmed3217]

I don't usually comment on videos but damn! Especially the 13:24 part was absolutely mind blowing! Great work!

[@jaewannies]

As changing my career from a Data Analyst to a Data Scientist, this video really helps me to brush up my calculus to learn advanced statistic models. It changes my future, thank you for interesting and awesome video !

[@magicandmagik]

LOL physics students be mad speed and velocity thrown around everywhere

[@nucle4rpenguins534]

I actually enjoy velocity problems lol

[@katienewman4743]

And the use of distance and not displacement, but this is an amazing video (and channel) lots of love!!!

[@henryg.8762]

yas

[@saiavinash7432]

@@katienewman4743 them vectors are a bitch.

[@chucksucks8640]

speed is the magnitude and velocity is the direction with magnitude.

[@spyrosmanolidis8516]

Thank you so much, your videos are perfect and are at an amazing pace. :)

[@burz4124]

I like the black background. Bc i see how stupid my face looks when I'm watching those kind of videos 😅

[@dunkeykung1162]

It's a whole army of stupids then mate, onward my fellow viewers!

[@jasperlee2238]

lmao

[@bananachocolatepistachio8696]

This is why I dislike the black background

[@realbignoob1886]

lol

[@salahshayah9864]

Best comment

[@michaelmcleary8566]

This is brilliant - I have struggled with calculus and at the age of 61, this has made it so clear! Today's students have some tremendous resources available to them.

[@okxa8857]

Thank you so much for this. Your explanation makes so much more sense than "decrement by one and multiply by the old power". It's far better than just regurgitating what teachers say.

[@nguyenducanh1462]

Math has never been this exciting to me !!! I'm so grateful that channel like this does exist on youtube Love from Vietnam

[@Mathze2]

I've always wondered whether it's possible to do an absolute perfect and flawless math instructional video. After watching this one, I know it is.

[@bctsnq]

Thanks! I never understood why derivative of x^2 = 2x. This video made it clear and it is set in my mind.

[@takeiteasy8847]

I am starting Calculus this semester. Thank you for your brilliant videos. They are truly inspiring and a joy to watch.

[@gold8783]

These videos are super good, this feels like something that I would have to pay for on an online course. I originally had no interest in calculus, but now I love it since you're always able to explain it with real world examples that are easy to visualize and understand. These make me really look forward to when I take calculus next year!

[@elizabethlugo869]

¡Este video es magnífico! Yo se que sería demasiado pedir, pero, necesitamos este trabajo en Español también, realmente tendrías un gran impacto en muchísimos estudiantes… Que genial material haces, muchas gracias.

[@natereinert1572]

Even though this video explains a concept in an amazingly beautiful way I still have to watch it multiple times to even kind of understand what he’s talking about

[@Mr.Obiology1981]

SOOOOOO glad I found your videos. I am a science teacher, and my math has gotten rusty over the years, so I am trying to dust it off improve. You are bringing so much clarity to intuitively difficult concepts. Thank you!

[@mr.ahmedfouda]

شكرًا

[@joshtipton7417]

6 minutes in and I've already learned more about rates of change than in my high school and college calculus classes combined

[@AryaDrottni]

I'm studying for my master's degree in theoretical chemistry and I love watching these videos. The "Essence of linear algebra" helped me to understand change of basis and linear transformations like no other video could. Thank you so much and keep up the hard work.

[@TruthBeTold.]

same for me tbh

[@tha1ne]

im an old man who has never taken calculus, this series is so helpful and easy to understand. THANK YOU!!

[@joeyvico]

This is the kind of tutorial that makes me love mathematics. Congratulations for the great video

[@vimalalwaysrocks]

Every student who says "Math is the toughest subject" really means that their Maths teacher did NOT do a good job teaching them math. Am one of those students! This video rekindled my interest in Math as an adult. I wish this existed during my school days

[@shadymello9146]

Or the ones that didn't practice enough... sincerely, A student of the latter kind

[@Mr1987atul]

The people with such deep knowledge will ask heavy salaries. Bcoz they know their worth. And schools cant afford such teachers. Which is why we get poor education from teachers with poor

[@MrCmon113]

Bullshit. Math IS the hardest subject, that's not the teacher's fault, but simply in the nature of the thing. You won't find a mathematician, who is not vastly above average intelligence and thinking about problems all the time.

[@thewonderer9551]

@@shadymello9146Its not only about practice. If a student doesn’t understand irrespective of practice for so many times & if a teacher isn’t good enough to make the student understand, the its not the fault of that student.

[@No-cg9kj]

No...most people are just really dumb when it comes to math lol.

[@DavidLDana]

Damn this is amazing. I have a calc test in a few days and I don't wanna rush any of this. It's to beautiful and I watch each episode like 5 times to really grasp the idea. I usually shrug at patreon but once i get my pay check i have to support. Thank you so much for helping me understand, but mainly, helping me appreciate math so much more.

[@noamsachner.5129]

I am 13 years old and I Would like to say thank you for this incredible series it has helped me a lot in my Studies of mathematics

[@kv-2450]

Mf is 13 and doing calculus

[@wellyesbutno9396]

ok

[@25nacesanleedaltrez58]

keep it up!

[@rabia1180]

beautiful! lots more cool stuff to unlock :)

[@3thanguy7]

OH MAN, THIS IS GONNA BE GOOD

[@bumdeedum8771]

Ethan Rojek You be right mate!

[@Aequorin628]

HELL YEAH! LET'S DO SOME CALCULUS!

[@umnikos]

Ethan Rojek I CAN'T WAIT UNTIL I BEGIN SOLVING THE 3x+1 PROBLEM USING CALCULUS!!

[@breckr1121]

Evan Bialo AWW FUCK YEAH CALCULUS

[@joraldthewizard2327]

kkk

[@DH-oj2ru]

It is astounding how much better of a teacher you are than my calc professor. Thank you for taking the time to make these videos, the level of intuition I have received from this series has been incredibly helpful. Maths are taught like shit in universities - I know this because watching your videos for an hour has given me a more in depth understanding of these concepts than my entire semester of calc 1. You sir are a legend. Anyone who is struggling with calculus should stop what they are doing and watch this series in full.

[@legendariersgaming]

I've never been so excited for a video series! :o

[@symbioticcoherence8435]

I have... for the last video series of 3Blue1Brown :)

[@Amaro27]

I'm currently taking Calc 3 and I still had questions about what limits and the derivative really meant. Calculus has such an amazing background, such complex implications of what we think about change and infinity, and yet many teachers just swift through it as if it were nothing. Thanks for doing these videos. Knowing that people are out there working for quality education is such a relief when regular class periods or teacher schedules aren't enough for engaging in specific topics and doubts :)

[@No-cg9kj]

Skill issue.

[@themisir]

Man, I am so grateful and thankful for your explanations. I've never felt enjoyment of learning something in last 4-5 years like this.

[@Cowmoo83]

Fun fact: The reason the displacement function is denoted by s(t) goes back to the German word for distance, "strecke" (despite the fact that this is the word for distance and not displacement).

[@LaureanoLuna]

Are you sure it doesn't come from the Latin Word "spatium"?

[@shrankai7285]

I am in pre-calc right now, and this is soooo helpful! Thank you

[@RelativisticInfinity]

how does this guy teach better than a professor of 23 years... i learnt now that the concepts i have built by practice in math could have taken a lot less effort and sleepless nights, and i might even have gotten better! keep up the great work, Grant!

[@ivanekkk1269]

"It´s kind of FLIRTING WITH THE paradox of change in an instant whithout ever needing to actually TOUCH IT " 3Blue1Brown That was funny :D

[@VTX_DePrinz]

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.

[@deedewald1707]

Well put !

[@GentleOddity]

I love this series! I am currently taking AP calc in high school, and this is the most sensical explanation of derivatives i've ever seen!

[@amirjutt0]

At 1:52 "But that guy already has another full time job in calculus" 😂

[@krzemienpasiasty3371]

Dziękuję polskiemu tłumaczowi za tłumaczenie serii tych filmów. I autorowi za tak dokładne wytłumaczenie tematu. Thanks for polish translator.

[@Saylor3561]

For a long time I’ve had a negative view of math. Like many kids in my class I said “I hate math”. A few years ago I tried changing my mindset-I was actively learning and investigating like some hacker trying to break a code. After changing my mindset not only did I beginning to appreciate math more, I also got better at it which made me like it even more lol. Next year I’ll be taking calculus in college and at the end of high school I got a distinguished certification for taking 5 math classes. Ngl I’m a little frightened by calculus-I’ve heard a lot about it-but I’m hoping that the admiration for learning and the joy solving a problem will get me through. Also of course being introduced to these concepts during the summer. I really like watching math videos and there are so many calculus based classes I can take after 2 semesters which is also why I’m exited. I finally get to meet the foe I’ve been anticipating and learn new exiting concepts (math in high school is a little slow or much repeated material)

[@kjekelle96]

0:00 intro 1:07 central example 2:20 velocity 3:26 change in time 4:32 ds/dt in the real world 7:13 tackling the paradox in pure math 9:44 the true derivative 13:11 take a step back 14:18 "instantaneous rate of change" 16:23 outtro

[@bclaus0]

Holy shit. I never had problems with derivatives and integrals but it's only now that I understand what the actual notation ds/dt means. Thank you so much!

[@OpPhilo03]

My English language very weak but still I am watching your video multiple time for deep understanding. I had watch this video 5 to 6 times but I not completely understand yet. I hope I understand deeply in 15 to 20 days.

[@reckingcrew2277]

As a lifelong English speaker, this also took me several watches to understand. The problem likely isn’t your English but the fact that we are trying to understand the math that it took humans hundreds of years to understand.

[@ibrahimsani9718]

thank you patreon,you just made me see the useful part of furthermaths,even though i live in Nigeria

[@moonchild5267]

I m in last year of my high school.,.and found this series....it's never too late... :)

[@science-y9209]

Sameeeeee!!!!

[@manlyman1393]

I'm just impressed someone was actually able to discover the concept of calculus. Took a genius.

[@MaxAider]

Dude I'm freaking 15 years old, I've just barely learned progressions, but this is so easy to understand! Fantastic work!

[@boraaddmath9167]

Bro...I envy u....I learn progression at 18 years old

[@Gutagi]

Im also 15, but dont find this “easy” to understand, after watching twice and constantly thinking about it for a day or so I can understand it...

[@Slijee]

@@Gutagi I’m 13 and this sounds fun as hell