# What they won’t teach you in calculus

3Blue1Brown
[Classical music] Picture yourself as an early calculus student
zooming in by a factor of 10 It doesn’t really look like a constant stretching or squishing, for one thing all of the outputs end up on the right positive side of things and as you zoom in closer and closer by 100x or by 1000 X It looks more and more like a small neighborhood of points around zero just gets collapsed into zero itself. And this is what it looks like for the derivative to be zero, the local behavior looks more and more like multiplying the whole number line by zero. It doesn’t have to completely collapse everything to a point at a particular zoom level. Instead it’s a matter of what the limiting behavior is as you zoom in closer and closer. It’s also instructive to take a look at the negative inputs here. Things start to feel a little cramped since they collide with where all the positive input values go, and this is one of the downsides of thinking of functions as transformations, but for derivatives, we only really care about the local behavior Anyway, what happens in a small range around a given input. Here, notice that the inputs in a little neighborhood around say negative two. They don’t just get stretched out – they also get flipped around. Specifically, the action on such a neighborhood looks more and more like multiplying by negative four the closer you zoom in this is what it looks like for the derivative of a function to be negative and I think you get the point. This is all well and good, but let’s see how this is actually useful in solving a problem a Friend of mine recently asked me a pretty fun question about the infinite fraction one plus one divided by one plus one divided by one plus one divided by one on and on and on and on and Clearly you watch math videos online So maybe you’ve seen this before but my friend’s question actually cuts to something that you might not have thought about before Relevant to the view of derivatives that we’re looking at here the typical way that you might Evaluate an expression like this is to set it equal to X and then notice that there’s a copy of the full fraction inside itself So you can replace that copy with another X and then just solve for X That is what you want is to find a fixed point of the function 1 plus 1 divided by X But here’s the thing there are actually two solutions for X two special numbers were one plus one divided by that number Gives you back the same thing One is the golden ratio phi Φ φ around 1.618
and the other is negative 0.618 which happens to be -1/φ. I like to call this other number phi’s little brother since just about any property that phi has, this number also has and this raises the question: ‘Would it be valid to say that that infinite fraction that we saw, is somehow also equal to phi’s little brother: -0.618?’ Maybe you initially say “,obviously not! Everything on the left hand side is positive. So how could it possibly equal a negative number?” Well first we should be clear about what we actually mean by an expression like this. One way that you could think about it, and it’s not the only way there’s freedom for choice here, is to imagine starting with some constant like 1 and then repeatedly applying the function 1 plus 1 divided by x and then asking what is this approach as you keep going? I mean certainly symbolically what you get looks more and more like our infinite fraction so maybe if you wanted to equal a number you should ask what this series of numbers approaches and If that’s your view of things,
but I want you to look at number two effective math and science learning cultivates curiosity. I love the word choice here. It’s not just that you should be curious in one moment It means creating a context where that curiosity is constantly growing. Just look at the infinite fraction example here It would be one thing if you were curious about why the numbers bounce around the way that they do, but hopefully the conclusion is not just to understand this one example I would want you to start looking at all sorts of other infinite expressions and wonder if there’s some fixed point phenomenon in them, or wonder where else this view of derivatives can be conceptually helpful Brilliant.org is a site where you can learn math and science topics through active problem-solving and if you go take a look I think you’ll agree that they really do adhere to these learning principles, coming from this video you would probably enjoy their “Calculus Done Right,” lessons and they also have many other courses in various math and science topics. Much of it you can check out for free, but they also have a subscription service that gives you access to all sorts of nice guided problems. Going to Brilliant.org/3B1B Lets them know that you came from this channel and it can also get you 20% off of their annual subscription.

## 100 thoughts on “What they won’t teach you in calculus”

1. 3Blue1Brown says:

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, as part 1 of 2 for an interesting application of complex functions with derivatives: Take a look! https://youtu.be/rB83DpBJQsE

2. axsk says:

A wild Julia appeared! @8:15

3. Steven Zinn says:

This is amazing

4. Rahul Shaw says:

Thanks for putting together the animations which I had to visualize myself. These animations makes sure, we are all on the same page and nobody is visualize something else while the point of discussion is something. I wonder if you are using any specific tool for generating these kind of animations or the generic ones.

5. dave p says:

Anyone on acid right now?

6. michael jonez says:

you get the Nobel prize!

7. Jason says:

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

8. Albamaria Rodrigues says:

5:53 got me

9. Kevin Luo says:

Geography.

10. Kamalesh Reddy Paluru says:

I always thought of derivatives as slopes and had this strange feeling I was missing something. Mapping almost made that intuition click.

11. jacktheninja says:

i just started calculus and only the first 6 minutes made sense :_:

12. isaiah mrman says:

and this video caused me to set the problem "what points on the gamma function converge?"

took me a year that one

13. Daniel Zuluaga says:

my god i loved this serie

14. Nathaniel Moto says:

all your videos make me feel dumb…

15. Mickelodian Surname says:

Ohh for hells sake its Fi… As in fly with no l … What is this fee?

16. ahtan2000 says:

Man… those graphics are world class…

17. michael steven says:

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

18. Tokaji Leo says:

for me this just confuses things and makes it more complicated., I learned calculus and used it but this video ..i just do not see what is the point of this. it is interesting and true but that's all.

19. Marko J says:

Everytime i thought I new almost enough about a topic, your videos showed me new insight and different viewpoint.

Your work helped me a lot getting a deeper understanding in mathematics.

20. Fiji Water says:

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

21. monogometr1c says:

Phi isn't pronounced as "fee", it's pronounced as "fai"

22. CartooNinja says:

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

23. vmkeys says:

Mapping input to output! I feel like this is one key I was missing on what derivatives do, and what all those rules and equations have in common. I hope I will find other keys in your videos and maybe, eventually, unlock this subject of calculus that I've beaten my head against for so long, only to have the it fly away again for lack of understanding.

24. Alejandro Quinche says:

This is one of my favorite calculus videos i think it gives a very nice understanding to a nice subject in differential equations and recursive problems, but that promise at the last minute to get more of it, it's a sour as it can get

25. larrybud says:

This is all Greek to me.

26. Anthony Alves says:

Thank you, I was having trouble with falling asleep

27. Andrei Sljusar says:

What does scale by 2/6 mean?

Thanks.

29. Vinícius Peixoto says:

This video is worse than calculus

30. Luis Martinez says:

What am I doing here in in Algebra 1😂

31. dudewaldo4 says:

"So, in the next video…"

🙁 pls giv

32. Rishabh Gupta says:

Loved your content always, Just amazing. Is there any series on Multivariate calculus too?

33. Timothy Morrison says:

34. Timothy Morrison says:

Why do they also not mention vortex math or hexagam casting

35. Bernt Sunde says:

Howcome my cat is fixed on a chairseat under the table after every dinner? Calculate that if ye'r able.

sir i need some help regarding this

37. Greg Jacques Lucifer's Jizz Gargler says:

teach me, daddy! best math tutorials online, bar none!

38. Riizen The Overcomer says:

So in other words children .. this is the mathematical process that occurs within your dna's x and y chromosomes as you're being born again in Christ Jesus aka "Phi" 🙏

39. Jesus’s Eternal Son says:

Zero space is a dark matter particle neutrino with magnetic 🧲 poles closed in from the outside space that is the location of the black hole.

So next tome you say zero think of a black hole 🕳.

40. Dave Ellis says:

As someone with minimal mathematical knowledge, is this an explanation of the definition of a function as given by Dirichlet, where every element of a given set (the input) is associated with a single element of a second set (the output)? Also thank you, wikipedia.

41. Hamid S says:

I got super sleepy right around 5 minutes Haha, complicated

42. Andrew Gonzalez says:

If you take every calculus course and watch the entirety of this video, you will retain the whole sum of human knowledge.

43. ThinkingOutLoud says:

13:39: “…kind of klunky..”
Uh, ya.
I’ve watched this video more than once and I see the point you’re making, but I think it would resonate more if this new visualization connected to some practical application rather than just being an alternative, kinda klunky, visualization.

44. Michael Darrow says:

Did you know, that 1+1/(1+1/(1+1/(…))) is the root to a polynomial?
x^2 – x – 1 = 0

45. Lycanroc Dusk says:

Y am I even watching this? I don't need X tra information

46. Andreas Raab says:

Ich wüsste nicht, was mir wer je beigebracht hat. Ferner hab ich die Infinitesimalrechnung selber nacherfunden als Bub auf
die Zur-Kenntnisnahme der Tatsache hin, dass in einem Fotobildband so komische Rechnungen als Dekor zu sehen waren.

Als ich den erst nur unvollständig angezeigten Cliptitel las, als nur ,,Was sie dir in der Infinitesimalrechnung..'' angezeigt war, ergänzte ich ihn mit ,,sagten'' und sogleich fügte meine Verärgerung ,, , hab' ich mir selbst gedacht.'' hinzu.

Dass das Tunwort ,,beibringen'' nahe dem Verbum ,,zufügen'' liegt; ist ein Dokument und einzig die Metrisierung
kann Thema einer Debatte sein, in die ich mich sehr zurückhaltend einmischen würde. Es mag sein,
dass der, dem etwas ,,beigebracht'' wird, die ihm etwas Beibringenden in just der kollektivierenden Anonymisierung zu Schemen wahrnimmt, mit der derjenige, der auf dem Operationstisch liegt, die Operateure im blendenden Gegenlicht sieht.
Sodass es nicht wundert, wenn sich im Titel des Clips hier just das ,,sie'' findet als die klassische Bezeichnung der Halter,
wenn das Lebewesen zum Lebewesen spricht — über ,,die''.

Ich gebe zu, dass ich mir nicht anders zu helfen wusste, als das ,,just'' hier zweimal zu setzen.

(Übrigens gibt es eine ganze Litanei der Bravour meiner autodidaktischen Eigenmacht, wobei ich nicht auf das Nacherfinden beschränkt bin. Meine Beschränkung ist stattdessen spürbar gegeben durch die Nachstellungen
derer, die ,,die'' sind; und die es soweit brachten, dass ihnen — den von Steuerabgaben verpflichteten Vollstreckern volkstümlicher Missgunst — Gutachter zur Hand waren, die befanden, dass ich ich ,,zu planvollen Handlungen kognitiv
nicht in der Lage'' sei — unter vielem anderen, dessen Zur-Kenntnisnahme mir erwartungsgemäß vorenthalten würde.)
Die Analysis und ihre topologischen Kinder mitsamt den postkartesischen Rangranglern der Raumstrukturkunde der
drolligerweise als als ,,Funktionalanalysis'' daherkommenden Lehre höchst exquisiter Witzelei über die Differenzierungen
der Unendlichkeit ist ein unendlich schönes Knetspiel der Anschauung, beinah die Schönheit selbst, welche

euch wie mir — und dem dies anzusprechen geistmächtigem Penrose, der die Adelung nicht verweigerte, anders als dereinst der Konrad —

erscheint als der Lack des Physischen. Faszinierend, dass diese Anschaulichkeit gegeben ist, welche eigentlich nur
das Accompagnato des Kartenlegens der b e i d e n Quantoren ist, dem Spiel, mit dem ich durch mich und mit mir
can be fooled all the time again.
Ich persönlich als ein Verehrer des Q u a r t e t t e s von 01Leerzeichen und — ,,…'' (All protoinfinities) bin — nicht
heute, ich bin vor allem am Dichten, wenngleich .. — bin beinahe ergriffen davon,
dass auf der Metaebene der Quantorensequenzierung das tatsächlich binäre Regime der Quantoren herrscht
im Medium formulierbarer Mengen — dazu sagt man in der Mathematik ,,Universum''.–
So als schwämme in ihm — erster Geist.
(Die Sprache ist also kein Lochstreifen, sondern eine Binärsequenz jeweils mit universumsmächtiger Modifikationsbandbreite intonierter Es-gibt und Foralls:) Die Infinitesimalrechnung war also der Aufbruch zu
dieser Sprache über die Rechthaber hinaus und verblieb nicht in den Pfoten des Schubsspiels.

Trotz erster Abneigung gegen diesen Beitrag, werde ich ihn mir vermutlich doch alsbald anschauen, so er wieder

angeboten wird.

47. Pepsi TaunTaun says:

Hmmmmm

Yes

The math here is made out of math

48. Jenny Guzmán. says:

Nombre en español 🙃🙃🙃🙃🙃

49. ThatHopefulArtist says:

Also Me: is extremely confused

50. Nightmare Court Pictures says:

This will help me make my redstone trapdoor.

51. Edgar Martinez says:

Because they don't like smart humans

52. Diana Riverjackson says:

After seeing this video I wanna stab my math teacher lol

53. Phos4us says:

Cam here just to flex my 95 in calculus.

54. TheAndJello says:

I have no idea what’s going on but I’m still watching

55. Abel Fiseha says:

Black ⚫ hole here we comeeeeeee🚀🚀🚀

56. Max Saenz says:

i dont even do maths

57. Main Culprit 718 says:

THE REASON WHY NEWTON IS GOD OF SCIENTIST

58. Pìnaki Chowdhury says:

Sir,can you please make a video on dirac delta functions introductory level

59. michel kluger says:

Another amazing video, thanks for the effort of bringing math in such an elegant and beautiful way

60. Jam Bos says:

Why do you pronounce the golden ratio as fee? It's pronounced phy like I.

61. Red X says:

What's interesting to me is if you replace the 1 in the infinite 1+(1/(1+(1/(1+… formula with other numbers.

If you plug in low negative numbers, it approaches zero, but if you put in -4, I'm getting a value approaching -2.

Interestingly enough, when I get the differences between steps, it ends up being the number of possibilities of two numbers chosen of the number of divisions in the formula plus 1…

-4=-4
-4+(-4/-4)=-3 (difference of 1)
-4+(-4/(-4+(-4/-4)))=-2⅔ (difference of ⅓)
-4+(-4/(-4+(-4/(-4+(-4/-4)))))=-2½ (difference of 1/6)

It gets chaotic when you plug in numbers between 3 and 4, and quickly convergent below -4 or above zero.

But exactly at -4 that pattern emerges.

62. Raymundo Wellington says:

I know one thing they will never teach you, and that thing is when you shoud stop NONSENSE like the Sign rule, there in operating Bananas and Apples, and by magic we have only Bananas !!

63. uwu Tunes says:

Wow, what an interesting way to present these ideas. And a pretty picture!

64. HN Tran says:

I would love to see a series on Essence of Probability and Statistics?

65. Pan Zielynsky says:

Im in middle school and dont know what are you Talking about, but i still loves to watch your videos. Keep it up👍

66. Austin Downing says:

Proof the earth is flat

67. Scυb Ŧσσ ᑕσσl says:

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: …

68. Paul Wattellier says:

Is that a pdf of that great video ?

69. AUSTIN SHOEMAKER says:

This video has the entire human knowledge besides calculus

70. Deepak S.m. says:

What they won't teach you in Calculus:

Most of them don't teach calculus really…

– What i think after watching these videos

71. Raghav Kumar says:

This video is too intelligent for YouTube…

72. Peterson Gomes says:

I understood nothing. Could someone explain to me?

73. Aryan Bhattacharjee says:

Like the like button if you like Physics!!!

74. Niels Ohlsen says:

IT'S THE SAME!

75. Bulldawg says:

I'm a big Grant Sanderson fan but typically can't keep up with his videos even with his excellent graphics … This is my deficiency and not a knock on him …

76. Samuel Kuhn says:

Never took a calculus class in my life and hate doing math. Yet here I am watching math videos for enjoyment 😄

77. Yeongheon Bae says:

14:54 어떤 인간이 저렇게 자막달았엌ㅋㅋㅋㅋㅋㅋ

78. cs127 says:

now watch the entire series again but this time, take a shot every time he says "tiny nudge dx"

79. Keane Jonathan says:

Can you also do an episode on the Gaussian Intgeral and change of variables in multiple integrals

The expression at 11:51 equalling approx. -0.38, would it be -0.382 to 3 sig fig? That is is the smaller number when dividing 1 in the PHI proportion.

81. Buehler1997 says:

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…

82. saketh pilli says:

me at 10 years old — "THIS IS NONSENSE"
me at 11 years old — "THIS MAKES SENSE"
me now, at 12 — "….. duh its obvious …. i learned it like what 2018…?"

83. Chy 75 says:

What does the integral of a function means when you look at the transformative view?

84. cosmos and physical laws says:

http://engineering-and-science.com/calculus-for-dummies.html

85. Naman Jain3102 says:

Which software u uses to make such satisfying animation?👍

86. Davide says:

still waiting bro

87. Balakrishnan Ganesan says:

Can someone send the next video's link

88. TheBlueRaven says:

One year has passed where is the next video in this series

89. Andrew Chan says:

I have a feeling that the education system doesn't want you to be smart but rather, a regurgitating, memorizing robot with no creativity and intuition.

90. IslandCave says:

At 11:48, you mentioned about -0.38, well I suspect its actually -(2 – phi), as phi + 0.38 is about 2, and it should be a value with some signifcance and not just a random value.

Also note that the reciprocal of phi (being about 0.62 (the negative of the phi's letter brother)) has exactly the same digits after the decimal part as phi. and so the reciprocal of phi + that 0.38 is about 1. Meaning that as we know the definition of the golden ratio, that 0.62 is the bigger part of the stick and the 0.38 is the small part of the stick and as we know then, the 0.62 / 0.38 is the golden ratio value once again.

Also note at 12:03 you mention about -2.62, now this is -phi – 1, which is also connected.

And I am sure that there are a lot more really neat interconnections between all these numbers then I can recall on the spot or even know of.

But the real question is, is there a formula that can go beyond e^(pi * i) = -1 or even e^(pi * i) + 1= 0 and even include phi in it?, now that would be Crazy Amazing.

91. Sebastian Fischer says:

This could also be extended to explain the integration by substitution or more general the transformation formula

92. Gabriel Johnson says:

This isn't really important for the video, but…

Why do people pronounce "pi" like "pie" and "phi" like "fee"??

93. Kenny Maccaferri says:

I think this video – is the one I would choose to help someone unfamiliar with Richard Feynman had a valid point when he said "i love mathematics but I hate (pure) mathematicians.". I've just read Steven Strogatz an applied mathematician and his book on Calculus and it is wonderful! He explains this same argument against invariably thinking about calculus in terms of y plotted against x in two pages, talking about the (y) calories per slice of raisin bread (x) at 200 per slice, and how much his daughter earned (y) working x hours in a local clothing shop at \$10 an hour. He said we tend to think of these as numbers. They are not. They are linear functions. He then goes on to describe the exact same technique of an infinite fraction as in this video, comparing it to a magician ….. Here this video, perfect for 3 blue 1 brown's target audience no doubt, is like hearing a music theorist describe Beethoven's Moonlight sonata in terms of change of frequency of vibration through sound over a series of episodic arpeggios over time. Maths is beautiful! This video is ugly. Keep up the good work!

94. Dom Dq says:

Phi is pronounced "Fie." Watching this video was painful with you constantly saying "Fee."

95. Li Ou says:

me wishing this was there when I first learned calculus…

96. Kenneth Jordan says:

Brillant

97. SpiderMonkey boss says:

I came expecting a lesson on the pre-World War I intranational French politics , and you have severely let me down, Steven.

98. Josh Navin-Graham says:

I made a program that finds phi when given any number
x = float(input("Put in any number "))

while x != 1.618033988749894848204586834:

x = 1+1/x

print(x)

while x == 1.618033988749894848204586834:

print("phi = ",x)

break

99. Samuel Romero says:

Anyone know what that song at the start of the video is called?

100. Ahmed Magdy says:

Beleive me your channel is tje bsst on youtube thumbs up 👍🏿