The other way to visualize derivatives | Chapter 12, Essence of calculus

[@fluiypj]

Now I just need a video entitled “what they teach you in calculus” and I’ll have the sum of human knowledge

[@dayzonmurray8880]

CartooNinja lmaoo fr

[@squibble311]

he has a calc series, so you have gained genius

[@masterdementer]

If you are studying in school then you will learn about them in senior high school, 12th Grade. And trust me it's not something you will want to learn interestingly. I'm still in 11th grade but I have done the basic calculus and it's hell of trouble to use in actual calculations.

[@zebulonpike9024]

Master Dementer High schools don’t necessarily teach calculus in 12th grade, it depends on the district almost entirely. I took calculus in the 11th grade, but I knew people who only ever took precalculus in high school(it was taught as trig and a bit of extra algebra). Calculus is also incredibly useful once you get past the basics. Especially in physics (which is what I’m doing rn), it’s an incredibly powerful tool to describe how things behave, allowing conclusions that would be totally impossible or incredibly difficult to demonstrate through algebra or geometry

[@pilgrimonthelongroad2875]

A = {things you learn in calculus} A ⋃ A'

[@scbl46]

Him: “Picture yourself as an early calculus student, about to begin your first course.” Me, an early calculus student about to begin my first course: ... [UPDATE]: 2 years later looking back at this video and the responses to my comment, my passion for math and physics has taken a bit of a dive but still there! To all the people who said such nice and encouraging things, thank you! People like you are what keep me going in today’s world. I passed Calc 3 with a 101% and Differential Equations with a 94%, onto PDE’s now and it’s going great so far!

[@NovaWarrior77]

Good luck! :)

[@scbl46]

NovaWarrior77 Thanks! It’s been pretty awesome since we started, I love math so much and I’m always itching to learn more about it. My brother is currently a junior in college studying physics and I’m thinking back to when I was in middle school and he was trying to teach me what a derivative was. So good to finally be able to understand and I’m looking forward to the knowledge I will acquire in my future, pursuing the same field as my brother.

[@NovaWarrior77]

@@scbl46 So cool! I passed calculus three in the spring. Higher math and ESPECIALLY physics take perseverance, but you'll get through it! Lean on the resources around you (including your brother) and don't hesitate to ask questions. P.S. I'm going into physics as well!

[@squibble311]

you have a cool username, maybe use ς or ℓ or ℂ in the username?

[@kanadetachibana4716]

How’s your course going now?

[@pipdang2468]

To be honest, this feels like a video about calculus for people who are way past calculus.

[@moulin3818]

To some degree it is true. Treating differential as an "operator" and studying its property do leads to more advanced topic of math.

[@stroys7061]

I’m 69, took differential calculus in 1974. This made perfect sense.

[@reman3000]

@@stroys7061 i just turned 18 and am just getting into calculus. hoping to be as fluent as that one day!

[@yaxiongzhao6640]

To understand how things are, better to see the backdrop of that thing

[@twicebittenthasme5545]

@@stroys7061 pre73 for me and while I understood the presentation, I also, would see the "sticking points" if the basics of math weren't firmly established. An analogy...asking an apprentice carpenter to build a spiral staircase despite the apprentice not having learned how to read a tape measure. And sadly, it seems knowing basic math or measurement is no longer a requisite to even pass elementary school. Some kind of "no child left behind" repercussion and/or aftermath which now carries over into real life! Power dropped at local supermarket and cashiers absolutely did not know how to make change for a dollar without a calculator!!! They were actually confounded by a decimal?!? The world is in serious trouble...

[@MarkChimes]

Man... I have a masters in maths and your videos still manage to blow my mind. What a beautiful way of looking at derivarives, and what an elegant application to that fractions problem!

[@janeshashah9186]

Can u teach me maths

[@besmart6990]

I can what are you studying

[@allthesesnitches4465]

How do you guys get the magnitude of derivative just with a pencil and a piece of paper and some formulas? No PC

[@mathematicsfanatic832]

@Aastha g Indian right?.

[@mathematicsfanatic832]

@Aastha g this is the problem of education system in India. Maybe we'll get to know what's happening better when we go to college

[@a-aronpre-sent1447]

People who create quality content like this for the world to feel in awe about are people who make the world a better place. It opened me to appreciate and see beauty where others see despair and agony-in maths.

[@davidbergaragonzalez5653]

I see the beauty and the despair an agony...

[@lyranrenegade]

well said

[@santiagoerroalvarez7955]

That "next video" tease at the end broke my heart. I was so excited by the idea of a 3B1B video on holomorphic functions and the jacobian determinant, only to discover that, a year and a half later, it still doesn't exist. It's sad that my first comment on one of your videos is just a lame upload request, so here you go: You have made me truly fall in love with math. Don't get me wrong, I've always liked the subject, this isn't one of those math redemption stories. But I used to like it in a much more different way, only appreciating the challenge that trying to arrive at a result poses. Your videos have totally changed my view of math, from symbols that obey certain rules and prove to be a useful toolkit, to something valuable in its own right. Something dynamic, endlessly explorable and, ultimately, alive. With your marvelous way of communicating, you have sparkled my curiosity and made me eager to learn, every day, a tiny bit more about math. And, for showing me the immense beauty this subject has to offer, I cannot thank you enough. Thank you for every moment of your time you have invested in this channel. You have changed my life.

[@vladturcuman7168]

kzbin.info/www/bejne/pmWZgoqvfbZ4bK8

[@NerdWithLaptop]

He uploaded a video on holomorphic functions earlier.

[@PietroMola-gz9dd]

@@NerdWithLaptop were?

[@bartuarik2424]

im still waiting on this

[@3blue1brown]

The brits among you yell at me, for how I say the letter "phi". But ask a Greek, they won't deny, there's something odd in saying "phi".

[@lucasng4712]

10/10

[@exod4]

3Blue1Brown dropping the hottest bars

[@KurtKremitzki]

If phi is phee, then pi must be pee, and I'm not ready to live in that world.

[@friedkeenan]

π is also pronounced "pee" just like our letter P

[@jerrynovotnik]

pi is truly pronounced pee in many languages like it would sound in Greek: kzbin.info/www/bejne/aGncpmSGe8hnaZom. Only English butchers it into pie..

[@alexdowad947]

...Just realized that 3Blue1Brown is the same guy who did the multi-variable calculus course for Khan Academy. That is probably the best math course on KA!

[@xkilla911]

i would appreciate a link to the video of the multi-variable calculus course

[@allthesesnitches4465]

@@adiabadic It's all about that fixed point. Haha

[@filipposaatkamp6337]

Same

[@theralhaljordan7337]

@@xkilla911 just search khan academy multivariable calc

[@howardbaxter2514]

What a great bonus for arguably the best type of calculus.

[@kjekelle96]

0:00 intro 1:35 the standard visual 2:22 the transformational view 5:38 application 9:45 graph vs transformation 13:32 the point 14:41 outro and sponsor

[@AJ-er9my]

'Picture yourself as an early calculus student, about to begin your first course.' Haha, that's me this upcoming school year. Watched through these videos on a whim and have really appreciated them. It's made calculus seem intuitive in a way that my friends who took it through the traditional school route haven't been able to garner. I really appreciate this and can't wait to start applying the concepts I learnt through this series!

[@AndrewGazz]

"Clearly you watch math videos online" Finally! Validation!

[@doge_69]

Same, this made me feel powerful!

[@shubhamvishwakarma3629]

You made me understood in 15 minutes, what my teachers failed to explain me over an year.I wish if I had teachers like you.

[@jeffreybonanno8982]

I think that, if I had been approached, or otherwise had been gifted an inquiry pertaining to your primary school teachers as well as, Junior High/Middle/High School level educational tutors and professors, and to what academic deficit they seem to have perpetrated in the their curriculums, again, based solely upon a perfunctory glance at your attempt at informal quick shorthand text style notation...might English and or Writing Comprehension have been one of your less favorite subjects, or is English perhaps a second or third language??? In no way is this meant to hurt your feelings whatsoever. I wouldn't have written it in a verbose and linguistically magnanimous manner of such simplistic vocabulary, yet elegantly oriented the verbiage for a slightly feigned complexity, and almost faux reflection and glittering glimmer of a spitshined surface. anywhere, I digress, and I would have benefited quite a piece, most assuredly, from some of the same teaching techniques and warm intonation of this style of technique and emotionally empathetic attitude integration. Some will never understand that "IT'S NEVER EASY TO TEACH ANYTHING TO SOMEONE THAT ISN'T WORTHY OF LEARNING IT!", and "IT'S NEVER HARD TO LEARN EVERYTHING FROM SOMEONE THAT IS WORTHY OF TEACHING IT!"

[@venra8920]

@@jeffreybonanno8982 man...get some help

[@henryrugg4971]

@@jeffreybonanno8982 try not getting high before opening this app next time Jeffrey

[@ana5tas1s]

Every time i get a notification for 3blue1brown i know my love for maths is about to be reignited

[@sarthakakhawat1893]

Ben Rowley Mee too

[@jas4768]

My friend, that flame should have never extinguished in the first place.

[@wirito]

bind like charmander holding up a little leaf on top of his tail under the rain...

[@n484l3iehugtil]

Why did I read that as "my love for maths is about to be reintegrated"

[@3blue1brown]

Instead of the follow-on I originally had in mind, which would extend these ideas to complex functions, the next video is on Divergence and Curl: kzbin.info/www/bejne/qHObZHemd6-Eqac

[@factzilla3071]

Just wondering, what program do you use to create these animations?

[@Thomas-vn6cr]

Congratulations on your million subscribers! : ).

[@KK10155]

wondering too, anyone?

[@hepa2406]

github.com/3b1b/manim

[@Yev371]

Will you touch bases on the Jacobian though?

[@NightmareCourtPictures]

This will help me make my redstone trapdoor.

[@ekrem_dincel]

No, it is more than calculus.

[@titanofchaos5917]

You need a PHD in Quantum Biology, Astro Philosophy, and Theoretical Algebra to do that.

[@ekrem_dincel]

@@titanofchaos5917 they are fine for a button.

[@technoultimategaming2999]

Differential calculus Linear algebra and Multivariable calculus is needed. Along with complex numbers, hyperpowers and hyperbolas. Ohh did I mention that you need to find a cure for corona and cancer too just as a tutorial?

[@Blue-hs9tv]

TitanOfChaos also biological chemistry, philosophy and graph theory

[@Astronomy487]

calculus teachers hate him!! learn how he graphed equations with this ONE SIMPLE TRICK

[@danielmelanson9531]

Astronomy487 lol

[@samsammich8465]

yeah it's still technically a graph that he used to explain his point it just happens to be in one dimension. He PLOTTED things on a line. he's like oh look at this emergent pattern, as he looks at what is essentially a 2d graph. and this sure as hell won't make vector fields easier

[@smort123]

What then happend shocked me

[@-danR]

You won't believe what the graphics app saw next...

[@AttilaAsztalos]

"One weird trick the Math Police doesn't want you to know..."

[@MathManMcGreal]

Love your insight on "conceptually lighter" topics like this coupled with your calm voice and chill music. It's like a soothing math meditation...

[@willyj3321]

I didn't even realize there was music until you pointed it out.

[@harryb7505]

Now that I have done calculus and watched this video, I now know everything there is to know in the universe

[@__-rs8kr]

Avatar checks out

[@santiagoargayo6409]

I dont think so m8

[@shenzou4778]

really? how could u describe the universe???

[@ooseven4696]

@@shenzou4778 you could actually equal the universe to 1 equalling a single entity(You really can't, this was a joke)

[@shenzou4778]

@@ooseven4696 lol, I think 42 is better choice, or 137.

[@algorithminc.8850]

I always look forward to notifications from this channel. The creator's love of mathematics is obvious, and he is extremely good at finding new ways to see and present it.

[@superj1e2z6]

You should do an Essence of Complex Analysis series.

[@Digvijay-dp5bk]

Yup, I also want a series in complex analysis , would you please do that!!

[@gargic651]

I really need it too.been struggling with that for a couple of weeks

[@liabraga4641]

That would be amazing

[@chobyriley417]

Definitely

[@BlueRock704]

Oh man, you're a hype-man.

[@akash3478]

If you know Grant Sanderson Your calculus must be great. I am a 16 yr old kid studying in 11th grade , Thank you so much !!! I not only understand derivatives and integrals and limits but also I can visualize them in my mind. Thus, it's really hard to forget them now. Thanks to you. I can't believe you give access to this series for free. Just know that you are doing a really great job. Also! Calculus is something which we learn in 12th grade. But as I was really excited because of the first video , I watched the whole series. I try to understand and visualize the reason behind each and every single derivative every single day since I have started watching the series. When my classmates see my notebook where I have practiced calculus , they tell me that I'm just wasting my time as we have to study the same in 12th grade anyway. But the intuition about this topic I have developed is indescribably amazing. Once again, Thank you.

[@aarjith2580]

wow awesome bro! what other great channels do you recommend, and wished that u must've discovered them earlier?

[@Maverick56912]

how you doing?

[@ir2001]

I don't usually comment on videos, but after exploring ur channel deep, everytime my notification bar pops up with those blues and brown, my passion to learn seems to get scaled by a huge factor, or in ur words that limit tends to infinity lol. The kind of visual intuitions you give really integrates the base of all these fascinating concepts strong in the spectators' mind, and I really wish I get a math teacher like you!!

[@iabervon]

I used to not care at all about new 3b1b videos, but that turned out to be an unstable fixed point.

[@nawdawg4300]

The music, the voice, and the animations are simply amazing. By far this is my favorite math channel on youtube! Oh and the Pi guys too! Keep up the great work!

[@jaday2246]

Currently watching this video about calculus instead of actually studying for my calculus exam tomorrow

[@aidenb3069]

What a mad lad

[@paulobatitay9362]

A little bit late, but how did it go?

[@StevenDinwiddie]

Same

[@UnQuacker]

@@StevenDinwiddie same :c

[@ishworshrestha3559]

Ok

[@ericpatterson8794]

I've taken calculus... Based on the title, if I watch this I'll have the total of all human knowledge!

[@joshuaabrams6355]

Ok. So his title should have been "What they won't teach you about calculus in calculus.

[@steliostoulis1875]

Hahahaha

[@avavaviv1]

clickbait de la clickbait

[@inversehyperbolictangent3955]

Well, technically, he doesn't claim he'll teach you what that 'what' is so you understand and know it fully, just that the 'what' will be mentioned in the video. So you may not gain knowledge and understanding of 'the total of human knowledge', but you may learn *what* 'the total of human knowledge outside of calculus' happens to include. The table of contents vs. the actual content of the book.

[@jadnajjarine6270]

Hahahah

[@buffplums]

I have always struggled with maths and oddly I now teach at a basic level. I still struggle with more advanced topics...to me this is advanced but your approach has taken me further with this visualisation approach which in fact has helped me grasp the subjects I now teach. I find that when a student is confused or can’t see a concept, I often find if I can visualise the problem then some of the students then get it. On the other hand. I get students who simply accept the principle first time and don’t like for me to over analyse the subject. Saying all that I pretty much get it up to 5 minutes in this video and then lose it but that’s progress ...thank you sir .

[@newmanmo2]

I use the word "best" sparingly, seeing as it's often used for people and structures that are at best average. Here, I am using it with utmost sincerity. Your channel is the best math channel. We are immensely grateful for your dedication.

[@veganath]

I concur!!

[@conoroneill8067]

I find it really interesting how two at first seemingly different questions can end up having a strong link - just the other day I was looking at the logistic map and its attractors, and was trying to find solutions at various inputs, and figure out which of those inputs were stable and which were unstable. I wish I'd seen this video before then - I was hopping between using calculus and using the graphical intuition, without a whole lot of connection between them, which was kind of frustrating. (I wish I had enough programming knowledge to try and map the logistic map in the way you did for the 1+1/x function here, so I could see it visually.) Actually, I'd really like to see some 3b1b videos on Chaos Theory, because its the kind of combination of visually accessible images and deep math that would work well in a 3b1b video.

[@alexpotts6520]

The logistic map isn't quite the same as 3b1b's graph. In this video, 3b1b just had an ordinary graph of a regular function and was using the graph to find fixed points. The logistic map is really a graph of different functions, a function of functions if you like - the x-co-ordinate is the parameter λ in the iteration z -> λz(1-z), and the y-co-ordinate shows the fixed points of that function for different values of the parameter. So it would be a bit trickier.

[@conoroneill8067]

Yeah, that's true. Still, for any given value λ in the 'stable' part of the logistic map, the principles that this video talks about apply - in the regions where there are oscillations of 2,4,8,... period cycles, then there also exist unstable 'repelled' cycles, or even a single point which will repeat, but points near that value will diverge away from it.

[@firefly618]

Ohh, that would be beautiful indeed if explained with 3B1B's magic. I remember trying my hand at coding various visualizations of the logistic map and the bifurcation diagram, ages ago, with early programming languages. (We didn't have Python or Mathematica back then, it was Turbo Pascal in MS DOS and a 640×400 monochromatic screen!)

[@Sonnentau1]

This looks like a problem you can solve with moddeling it like a closed loop system. You will find all stable and unstable points by finding the poles. For continous systems: If there realpart of the pole is smaller than zero it is stable. For discrete systems: If the realpart of the pole is within the unit circle it is stable

[@GerBessa]

pyropulse At this level of learning, being condescendant becomes very much counterproductive to anything. If it was a mechanism aquired while preparing a competitive exam, I understand its origin, but when you move into the world of advanced math and problem solving, there is always someone with a better understanding of how two separate domains interact and asking for insight or help isn't a proof of weakness, but an efficient way to progress. TLDR : You're smart enough to understand what he asks for, so don't be a douche.

[@mike814031]

this is by a long shot the best explained version I've ever seen, well done! i love how well explained these videos are. it makes me wonder why all the teachers I've seen have never bothered to explain things so clear

[@cayezara8110]

Wow! I am a graduates of Engineering and I am so blown up by your illustration and explanation. You are gifted.

[@evasuser]

and yet another invaluable video, not only because it presents the derivatives in a new fashion, which is awesome in its own right, but because it inspires us to think that maths is a useful tool which helps to gain intuition on how a process or phaenomenon behaves. And one more thing, it makes maths a lot more attractive and easier.

[@minerscale]

You monster, I was just about to go to bed.

[@Someone-cr8cj]

Miner Scale lol

[@matthew55793]

Suffer

[@xnoreq]

You should seek professional help.

[@rigille]

What's your timezone?

[@sundaralingams8083]

Same here. 😂

[@fiveoneecho]

I’m rewatching all of these videos after having taken calculus, and they make so much more sense.... I’m being awakened to so many cool concepts that make so much sense!

[@shreyasraut6224]

I am out of words to describe how good you are at explaining difficult concepts. Since the time I have started following your channel, my perspective of Math has changed.. Thanks a ton... Super awesome video...

[@jyotishkaraychoudhury4762]

Your videos are a good reason behind my love for math.

[@stevematson4808]

I greatly appreciate your presentation on the subject of calculus. One of my frustrations in learning this subject is that professors (usually older theoretical mathematicians) will speak of calculus in almost mystical and vague terms. This can be very discouraging to beginners. I have actually learned more by avoiding bad math teachers.

[@destroctiveblade843]

I guess one of my favorite moments this year is when studying linear algebra , we were being tested by this professor who saw that most of us were lacking in the visual department and wrote the name of your channel on the board , and I was like "you know 3b1b ?" We both started chatting about how great your channel is , that was genually a good surprise

[@soyoltoi]

Lone Wolf The professor could have explained it to them and then tested them only to realized that his/her explanations were lacking visually, so he/she referred to a KZbin creator who is good with visuals. It's too easy to blame teachers.

[@aayush_dutt]

Eagerly waiting for the Jacobian explanation!

[@masonno4468]

Aayush Dutt why wait? Pretty sure he already did one for khan academy.

[@usfghost]

Love your profile picture

[@523101997]

yeah I remember him doing one

[@gentbar7296]

Aayush Dutt i just call it Pancho or Jose

[@soreff]

This _is_ the Jacobian explanation. The Jacobian determinant is the generalization of exactly this how-much-space-in-the-input-maps-to-how-much-space-in-the-output question to multiple dimensions.

[@christianherrera9363]

WHO DOESN'T LIKE THESE VIDEOS?! I had very good professors for calculus but you simply make it easier and more intuitive to think about the math with all this animation. I am literally watching this videos out pure entertainment.

[@Thee_Sinner]

This couldn’t have come at a better time. I start calculus next term and it will be the first maths class I’ve taken in 6 years..

[@heckler73]

Don't worry. You'll do well, especially with channels like this to accompany you. I would also recommend PatrickJMT's channel for more "grunt work" examples and assistance in comprehension of the stuff you will be learning.

[@Cody.Schrank]

If you seriously haven't done any math in 6 years make sure you brush up on your trigonometry. You'll need it.

[@Thee_Sinner]

Cody Schrank I audited precalc this term, but didn’t really put too much effort toward it since I was enrolled full time on top of that. I now have 2 trig books to use over summer tho!

[@quietsamurai1998]

Cody Schrank Somewhat related story: After I learned that I would be able to skip Differential Calculus using AP credit, I chose not to take a math course in my first semester of freshman year. I took Integral Calculus in my second semester, and I had already forgotten basically all of trigonometry. It's quite alarming just how much you can forget in < 8 months.

[@wengeance8962]

Hey!! I'm in the exact situation too!

[@loremipsum7513]

Oh god i love u 3b1b. I'm currently 3rd year cs student. I didn't really dive into math anymore. But your videos and the way u explain is just amazing. So good it makes me actually watch the whole thing just for the sake of enjoyment.

[@ThePharphis]

take an elective math! Some obvious USEFUL choices for those who don't remember/know calculus enough to take multivariable or complex or real analysis: set theory linear algebra

[@loremipsum7513]

@ThePharphis. fortunately I do get linear algebra as a compulsory courses :D

[@ThePharphis]

do set theory then ;) or more discrete math if you took less than is available! or maybe just upper year courses if you have the prerequisites like abstract algebra or real analysis... probably need a few calc courses for those, though.

[@chasefuller8496]

These videos are works of art and I love you for putting the time and effort into them.

[@meddelhed]

Your pronunciation of "phi" fills my heart with joy!

[@shivraj6664]

boi gotta flex those conceptual muscles

[@shmendusel]

weird flex but okay

[@syamantakbharti3463]

@Diego Marra XD

[@kiran0511]

One of the best channels on KZbin...wish KZbin would handout awards for channels with high content quality as well instead of just viewer numbers...

[@SidewinderScience]

your videos are so good I've subscribed twice. ;)

[@3blue1brown]

Haha, thanks Cody!

[@abcxyz6606]

you've actually unsubscribed since you hit the subscribe button twice. i've only hit the subscribe button once so i'm still a subscriber

[@roaringdragon2628]

surprisingly underrated comment.

[@dcn584]

mow your lawn

[@-danR]

dcn ha ha. Only Cody fanatics will get that.

[@prestonlui6451]

Being an actuarial science student, I am shocked that there is such a special way to see derivative and calculus. I really wish I had known this approach before.

[@ericransom4129]

The reason behind the ellipse 10:25 can be explained by the Dual Steiner Conic, because f(x) = 1 + 1/x makes a projective map (it preserves cross ratio) that sends one line to the other, and the point of intersection of the lines is not sent to itself (in this case f(∞) = 1 and not ∞) gives us that the line Af(A) is tangent to a conic (it is an ellipse probably because the lines are parallel).

[@erawanpencil]

When do you think the ellipse would be a perfect circle? Just by visual intuition, I think he should have just moved the two parallel lines closer together so everything was square, horizontally and vertically... i.e. it seems like the ellipse is just an artefact of him making the input and output lines too far apart, but I could be wrong. You say 1 + 1/x preserves cross-ratio, isn't that also true of human vision in the most general sense? I thought I remember that in projective geometry, or to make realistic art, the way in which stuff gets smaller towards the vanishing point/point at infinity is when the signed cross-ratio is preserved at -1. There's an article somewhere about how iterated projective harmonic conjugates approach the golden ratio, which seems like it must be related here but it's over my head.

[@hibye2627]

8:17 Start at any Number. *Inputs 0*

[@bentoomey15]

Also, inputs -1, gets 0, inputs 0.... Or inputs -1/2, gets -1, inputs -1, gets 0, inputs 0.... Or inputs -2/3, gets -1/2, inputs -1/2, gets -1, inputs -1, gets 0, inputs 0.... Or inputs -3/5, gets -2/3, inputs -2/3, gets -1/2, inputs -1/2, gets -1, inputs -1, gets 0, inputs 0.... Or inputs -5/8, gets -3/5, inputs -3/5, gets -2/3, inputs -2/3, gets -1/2, inputs -1/2, gets -1, inputs -1, gets 0, inputs 0.... Or inputs -8/13, gets -5/8, inputs -5/8, gets -3/5, inputs -3/5, gets -2/3, inputs -2/3, gets -1/2, inputs -1/2, gets -1, inputs -1, gets 0, inputs 0.... Proposition. Let F_n be the nth Fibonacci number, with F_0 = 0, F_1 = 1. Then the rational number -F_n / F_(n+1) "breaks" (by division by zero) the dynamical system f(x) = 1 + 1/x on the (n+1)st iteration. Proof. Note -0/1 = 0, -1/1 = -1. Suppose -F_(n-1)/F_n breaks the dynamical system after n iterations. Then 1 + 1/(-F_n/F_(n+1)) = 1 - F_(n+1)/F_n = (F_n - F_(n+1))/F_n = -F_(n-1)/F_n, so by the principal of mathematical induction, the proposition holds.

[@nanigopalsaha2408]

@@bentoomey15 Nice

[@stuartdparnell]

@@bentoomey15 someone needs to write an essay on the phibonacci function and its variant input possibilities... Eg e, log e, pi, -1, 0, primes, etc

[@JonathanLidbeck]

Math calls your bluff. 0 --> 1+1/0 = ∞ --> 1+1/∞ = 1 --> 1+1/1 = 2 --> 1+1/2 = 3/2 --> ... (converges to phi as expected)

[@masterspark9880]

@@JonathanLidbeck 1/0 isn’t infinity

[@Scripter-bx6zy]

What a wonderful way to think about functions! I have never visualized them like this before It does indeed seem like it could make a lot of problems and ideas easier to grasp

[@shokry1337]

Your Essence of Calculus series reignited my flames of love for Mathematics. Thank you, truly.

[@fationr.7272]

your channel is gold, truly GOLD. I salute you.

[@NinjaAdorable]

Engineering Masters student. Couldn't really wrap my head around this visualization ! #whatislife

[@Buehler1997]

these kind of videos in which you stop to understand things after 2 minutes but still you watch it until the end because it's very beautifully made and instead of thinking about math you start to think about how these beautiful animations could be done...

[@mattRRgraham1996]

Well the reason they don't teach the derivative like that is because for people who struggle with math more than others (which is totally okay!) or are new to the subject, a graph is the simplest way to demonstrate what the derivative does. Now sure, this is still worth knowing as it broadens your understanding of the derivative and if one way of understanding the concept is a little muddy, perhaps this will be an easier way for a student. I say this not to discredit the video, I love this channel and this video! But I conjecture that this might be a bit tricky for a new student to the subject to grasp in comparison with the usual way involving plotting some nicely function/polynomial. I just finished my math degree (still have a year left though, taking some grad classes in Analysis and Topology and a research seminar thing), and one of the sequences I did this year was one in Real Analysis (Intro to Analysis followed up with a course in Metric Spaces). It was cool seeing the concepts I learned in those courses come up in the video and be so well illustrated. Another course I took this past school year (actually finished two weeks ago on the 9th) was a course in Nonlinear Dynamics and Chaos Theory. It was cool seeing that little iterated map thing at the end there and discussion of fixed points :) I'm also pretty sure I saw that example (or one like it) one day in my Metric Spaces class when we were going over some fixed point theorems, but nonetheless it's cool stuff and I loved this video, thanks for making it!!!

[@mannyheffley9551]

Maybe they should offer a choice to contine with the course after an introductory class.

[@al1b219]

Superbe vidéo comme d'habitude : 15 minutes de vidéo => 15 jours de réflexions sur le sujet !!! Merci!

[@3blue1brown]

Ah, oui, c'est la raison pour laquelle je ne pas publier plus fréquemment ;)

[@LeEnnyFace]

Wait, are you french?

[@constantijndekker8343]

I am not French but shouldn’t it be ‘ne publie pas?’ It doesn’t matter, calculus is more interesting anyway!

[@al1b219]

Oui, exact mais le français est une langue complexe, tout le monde fait des fautes, surtout les français (; Pour les francophones matheux je conseille vivement la chaîne "El Jj" !

[@constantijndekker8343]

I am proud, I haven’t studied French in years but I think I know what you’re saying :) French is a difficult language. But I think Latin is much more difficult!

[@benwinstanleymusic]

I love how you can describe how you don't like the way cobweb diagrams are taught, while giving probably the best explanation I've heard of them so far

[@Earthcomputer]

Intesting. Is there also a similar way to think of integration?

[@zairaner1489]

I can't think of an immediate analogon, but this can at least help you understand the substitution rule

[@inversehyperbolictangent3955]

Here he is just relating two functions via parallel axes (rather than perpendicular, as in Cartesian grid). One function is just the input coordinate, c(x) = x, and the other function is just the derivative of some function under question (say, f(x) = ), which in this case would be f'(x) = d / dx. The transformation visualization in this case just goes between c(x) to f'(x). For integration, you could just keep the first function c(x) and relate it to F(x) = Integral( f(x) ) = Integral( ). To deal with the constant of integration (typically called C), you could either set it to C = 0, or change it and play with it by shifting the bottom axis left or right (as the above video did when showing 1/x vs. (1/x + 1) ). Really, this transformation visualization is just another way of showing/visualizing the relation between two functions (with the first function typically just 'x' without any modification). So, it can be used to show any transformation from one function to another, including x to the integral of f(x).

[@iabervon]

One property of the derivative that makes this video nice is that it doesn't matter where on the number line the region ends up. When we zoom in on the output, we don't have to know where on the number line the box is. The integral doesn't have this property, so you can't "think locally", at least at that part of the animation, and get the right answer. You also can't think locally at the input side, but that's a bit less obvious.

[@christophercampbell6906]

Yeah! I think of integration as a transformation into R^1, at least for definite integrals. This helps in analysis when you are considering the ordering properties of integrals. For indefinite integrals, just fix that idea of transformation to R^1, but for some new function that gives the values of this transformation.

[@acykablyatley]

if I understand this method correctly, as the functional inverse of a derivative; choosing an even distribution of points on the output line and mapping them back to the inputs, plus some constant. in the video it would look like points on the bottom line rising back to the top, but their concentrations would follow the same rules as the graphs; zeroes in f are the maxima and minima of the intergral of f, the most dense intervals, maxima and minima of f are the inflection points of the integral of f, the most sparse intervals.

[@MuzikaLov]

This is a really beautiful portrayal of the beauty of math and how it follows the same laws intrinsically as we do.

[@52Warlock]

Just want to say thanks for the great videos. My "Aha!" moment came in my first term of second year Naval Architecture and Marine Engineering. After a year of plodding through Calculus with no value added information ( e.g. why it's worth while ), the prof in the NavArch class began with a description of how one would calculate the waterline of a yet to be built vessel. After much expounding, he mentioned - almost in passing - that we must calculate the 'area under the curve' to determine volume. I exclaimed and got called out for my outburst. I said simply, "If I'd known this on the first day of Calc - I would have breezed through." Again - thanks.

[@52Warlock]

The curve defined by the underwater section of a hull. Take a boat - any boat - and flip it over. Then slice the boat from side to side like a loaf of bread. Each of these slices will have a unique curve that can be defined by a set of equations. These equations, in turn, will tell you where the waterline should be.

[@52Warlock]

Let's give this a try. Take a slice of bread with the top crust up. This represents a slice of the hull. Now take a very thin slice off the top at. This slice has a depth ( fore and aft ) and it has a curve to the top of the crust. If we use calculus we can determine the AREA of the front of the slice of bread. If we then multiply that AREA times the DEPTH, we get a volume of that small piece of bread. Now we know how much water will be displaced by the piece of bread if we dunk it. If we know the mass of the displaced water and compare it to the mass of the piece of bread before slicing, we can determine if the bread will float or sink at this arbitrary water line. Repeat the process until the mass of the bread (boat) is equal to the mass of the displaced water - that line/point is where the boat will float.

[@ryPish]

They won't teach you how to tie your shoes with a single hand, or what money actually is, or even what laws there are. For everything else, calculus can help.

[@Alex-vh4tb]

Ry P Yeah man Ive always wondered what laws there are.

[@DasMerkel95]

You can tie your shoes with a single hand?!

[@andriod8014]

Ry P learning to tie your shoe with one hand is not mandatory and needed, plus what class will teach that? About money, in some states (U.S) they are economic classes in HS which teaches about money.

[@glum_hippo]

Tie one shoe with each hand, save time

[@jamesmnguyen]

Money is a tool in economics.

[@Aurora_FGC]

I don't need to picture myself as an early calculus student because I am! I'm starting my first calc class after this summer and I'm really excited!!

[@mike814031]

this is my favorite channel for getting clear easy to understand explanations, I love how they explain it so well

[@XanderOwen24]

I've actually already taken Calc I and Calc II, and I'm pretty sure I saw this video over a year ago, but this video just clarified the concept of stable and unstable equilibrium points from my ODE class for me. Very interesting stuff

[@surodeepspace]

I don’t mind clickbaits from this channel

[@quietsamurai1998]

I don't think this is really even clickbait, because I've never heard of any calculus course that teaches this way of visualization.

[@franzluggin398]

Quietsamurai98 It nonetheless uses rhetorical tricks to push you into a certain emotional state ("*They* won't teach you but *I* will!" - Strong "us-vs-them" mentality) and conveys a certain urgency, as if whatever this video has to say could help you improve drastically (so you should watch it, asap!) _against the wish or design of whoever is in charge of "them"._ Phrasing the title this way rules out the possibility that other people try their best to instil the basics of university-level maths into students with wildly different starting knowledge. People who simply never considered teaching derivatives this way because they personally do not find it helpful or easier to imagine than the standard approach. Instead, the title implicitly conveys an intent on behalf of the unnamed others, "them", to withhold this information from you. This, in turn, gives an impression of deceitfulness, as there is really no good reason to withhold said info. That said, the phrase IS a meme at this point, and I have no doubt 3B1B meant it only as a joke. I don't have any problem with him using this video title, in fact. But even if only meant as a joke, it still remains clickbait, even if cheesy one delivered with a sly grin instead of a straight face.

[@me444444444444]

I disagree, I don't think 3blue1brown is implying that your school teachers or the education system as whole purposely wants to keep you bad at math. I see it as either a legitimate critic of how math is taught in most schools, or a reassurance to those who took a lot of mathematics in school but still don't grasp the "geometric intuition" behind it. You're right that its pretty clickbait-y, but at what point does clickbait simply become copy?

[@Wecoc1]

It's not clickbait if it doesn't disappoint

[@franzluggin398]

As said, I agree with your sentiment that 3B1B probably doesn't believe there is any evil intent here. I don't want to convince you that he did any wrong in labelling his video as he did. My point is that even though he used a phrase that is likely more often laughed at than taken at face value within our social circles, it is still a rhetorical tool most often employed within the context of us-vs-them polemics, and that usage is going to colour the viewers' impression of this video when deciding whether to watch it or not. Consider it from this perspective: A video's title is basically a short advertisement to watch it. And I would say that any product that doesn't advertise itself by showing its own good sides, but instead everyone else's bad ones, is being needlessly aggressive in its advertisement. And this title does not even allude to anything at all in the video except the fact that you won't get that content anywhere else, i.e. _everyone else is bad_ and not _I am good,_ as well as the general topic (calculus). In this case, it doesn't matter. I know 3B1B's videos, I would watch just about any video he uploads and it is very possible that the title was just a joking exaggeration. And I'm sure other viewers like this channel just as much and would think similarly. But it's still clickbait for me because if some other channel with more questionable video quality and intentions did it, it would feel like an annoying practice to grab attention.

[@satyajithbommanaboyana5596]

Human brain is truly so malleable. It is weird (& fascinating) to myself to note the following point: I watched this video right after it was posted and thought it is another perspective of understanding a derivative - as a "localized" scaling factor when f acts. I shoved it into the back of my mind and moved on with my learning process. I watched the same video today with a much better realization and the ability to make connections with many concepts of applied mathematics (the beautiful creation by mankind). This interpretation of derivatives (the process of taking derivatives as an action of a linear operator) is the basis of Frechet/Gateaux derivatives, contraction mappings, fixed point theory, existence/uniqueness theorems in differential equations, variational calculus, variational methods, Newton-based methods in numerical analysis, optimal control problems and optimization theory, in ML/DL and so many more concepts to list that I learnt after watching this video. I am amazed(surprised?) by the fact that my brain is able to notice and recognize this interpretation of derivative popping everywhere in applied math. I am assuming that 3B1B also made this video after encountering this interpretation a many times in various math topics.

[@mayabalaji1020]

Your videos are absolutely beautiful and provide different ways of thinking about mathematics. Please post more!

[@akashchandrabehera7667]

I had this intuition when I was studying a chapter in my school course book on Maximum and minimum values of a function. It's very satisfying to see that my realization was indeed true. Thank you for your video, sir.

[@ronaldkubiak5667]

I watch these videos over and over. They are the best thing on KZbin

[@yuvaldolev7969]

All of that was delightful as usual, but the thing I liked the most was finding out that I am not alone in calling -1/phi by the name "phi's little brother"

[@KematianGaming]

my brain already exploded before the first minute ended

[@gauravsingh3007]

I want to look inside an open brain xD

[@1.4142]

I tried to picture myself as a calculus student, and by 0:06 I was unconscious.

[@aasrithchandu7680]

I've just finished the entire series!! Just loved it.. Probably these are the most interesting videos I've ever watched on KZbin..!!!

[@karthikrambhatla7465]

Thanks a lot for explaining stable and unstable points with respect to derivatives.. this is so clear and amazed me..

[@klumpeet]

Beautiful video. Very relevant to probability, both in representation of density function with point samples and how mappings transform that density.

[@stuntchild]

The correlation this has to the golden ratio, or Fibonacci sequence theory, is quite incredible. This draws a whole new concept to me on what "the building blocks of life" coined phrase means. 1 : 1.618 ratio 10:00 - Look at the spiral shape he's drawing. 2:21 - Also, the wave frequency-like result is quite fascinating as well at this part of the demonstration. I can see the matrix.

[@lalatdas]

The best explaination of calculus I have ever seen. You spellbound everytime you post a video. I always wonder how come you are getting these intuition about these math concepts.. you must write a book describing all this. :-)

[@vishwas425]

0:20 'a few nice *aha* moments' 😝 That was sooo soft

[@specific_pseudonym]

I jumped into the "advanced topics in math" class at my college a couple years ago...it was about discrete dynamical systems; a topic that the professor's PhD thesis was centered around. Without a single doubt, despite struggling with the class because I had to get an exception for my lack of a proofs class before taking the highest-numbered math class available, it was my favorite math class *ever.* By FAR. It lent to me an understanding of not just how calculus works, but also how it was derived, and how it connects to physics (my major). It is rarely mentioned (probably a product of how it is not often offered to undergrads) but it's just...cool as hell. Thanks for doing a video on it :)

[@AvaloniA_]

I am a Korean high school student studying calculus. I really enjoyed the video, and it was very informative. 😁

[@ApolloGorillaTag]

I had a similar problem to the one you talked about on a math team test and I got two solutions, I wrote both down and it was marked wrong :(

[@chadchampion7985]

L

[@jabbahatt8082]

W for trying

[@abdalrahmanabyad2936]

You should be considered the “math teacher GOAT” Your explanation is on another level

[@hey7328]

mapping point from one number line (the x-axis) to another number line (the y-axis) is exactly what a graph is. the visual for derivative is exactly the same as the limit definition for a derivative

[@romanbykov5922]

the fact of saying "fee" instead of weird "fye" alone is worth a huge thumb-up!

[@TripleIProductions]

14:25 - This visualization made me realize why many types of graphs ( in this case f(x)=1/x which goes through point 1,1) never end and are therefore infinite. It is because they are based on the lines from a normal coordinate system, and they never end. For the example, it is based on the line right above the X axis (the y=1 line), which never ends. I'm very impressed by these visual representations. Thanks!

[@TripleIProductions]

And I know it wasn't even the point of the video, but it was still helpful!

[@jerrodshack7610]

With the combined knowledge of calculus class and this video, I know everything there is to know

[@emperorjustinianIII4403]

That lovely IPython makes me understand math instantly in contrary to long formulas

[@gareththompson2708]

Because I watch math videos on the internet does NOT mean I know a lot of math. It just means that I find math interesting. I'm not even close to taking a calculus level course, and yet I find your videos on calculus fascinating (and, for the most part, totally beyond me).

[@navjotsingh2251]

Honestly the main prerequisites to calculus is just trigonometry, logarithms, exponentials and just knowing how to graph functions.

[@kalebbruwer]

10:38 Can we just pause for a moment and appreciate that that's a perfect circle?

[@gdibble]

Great video, thanks for taking the time to share this 👍

[@felixszopos-papp1478]

If only this was the way every teacher taught. From your videos i'm starting to understand a lot of concepts our teachers failed to elaborate on. Great work! Thank you!

[@Brandon-dj9cq]

I don't even know the word calculus my brain: interestinggg....

[@rayyan21d]

Dang that's me about 3 years ago

[@panosr2135]

1.5 years until I start university and I understood everything, thats a sign of how awesome your videos are.

[@ShubhamMishra-ed7mb]

This is the most beautiful way of explaining differential among i have ever seen yet.

[@pathpathni568]

Only the ogs remember the original title: They Wont Teach You This In A Calculus Class

[@User-ot6jd]

That is a universal truth.

[@kjl3080]

I misread that as ln(A)

[@divadugar4695]

These videos are brilliant Period

[@johnchessant3012]

12:02 In LaTeX, you can write |{-2.62}| to avoid that awkward space between the minus sign and the number 2.62.

[@raunakdas4646]

I was banging my head in Differential Equations and here i learn there’s also Differential Geometry ahead.

[@alexandertownsend3291]

There is also differential topology.

[@berkakipek8991]

Thank you this will give me another perspective to gradients of a function.

[@wl50451]

This is a video that explains the "golden ratio" so clearly I have never seen before. Thank you, Grant!

[@fauxpassant]

Honestly, you remind me of Kekule, the chemist who dreamed of a snake biting its tail and realised/concluded that benzene is a ring. Your videos on visualising light houses to find some infinite sum or product, and "alternative views" of topics, make Mathematics look like something that can be a lot easier understood just by "simple visualisation". However, the reason why a large number of people still don't understand Mathematics is because aside from you, no one (or very few people) can actually think of the exact same "analogy" as you. So keep dreaming, 3blue1brown, bearing in mind that no one will ever arrive at the same analogies as you. At the same time, I must say you've done a great job in making all these videos. Your visualisations are really weird and, to me, extremely unrelatable, but at the same time, I believe it's going to help someone who has gone through a lot in Mathematics not understanding Calculus. When they discover your videos, it may finally light up the spark in their minds. Keep working, 3blue1brown! Even though your analogies are really unrelatable at times, I really look forward to your videos, because they remind me to keep an open mind in teaching. I used to do Chemistry, but now I teach a few students Calculus and Linear Algebra during my spare time, and your videos have helped me in explaining some key ideas to them. Thanks a lot!