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'

[@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.

[@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.

[@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. 😂

[@NinjaAdorable]

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

[@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.

[@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

[@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 :)

[@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.

[@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.

[@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. :-)

[@AvaloniA_]

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

[@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!!

[@antiolrachmor]

I wish all math classes were like this

[@blugaledoh2669]

lactha time

[@snakeoil7089]

Hmmmmm Yes The *math* here is made out of *math*

[@erik9817]

Cool to see the part around 12:51 about unstable and stable fixed points, we learned about them a bit in a numerical methods class.

[@berkakipek8991]

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

[@arg9559]

6:00 How marvelous the ellipse(?) is

[@MsAlarman]

It is just mind blowing how this can be applied to explain human behavior in social sciences! So indebted to you for showing us the limitations of geometry.

[@sarahp6512]

This is a great concept for understanding propagation of uncertainty. If a variable x in a function has an uncertainty, you can find the uncertainty in the output of the function by taking the derivative. With this visualization, that uncertainty is some small region around x. The derivative tells you how that uncertainty is stretched or squished as it goes through the function, giving you the new uncertainty of the output. Typically I'd visualize this graphically by taking the region around x, mapping it onto the curve, and then reflecting it over to the y axis, using the y component of the slope to determine the width of the new error bars. I think they both work equally well, but it's nice to have multiple visualization tools to pick from.

[@jimmylander2089]

Imagine yourself never having taken calculus and now finding out that you actually want to do stuff that involves calculus and you gotta self-teach yourself calculus. I personally do not have to imagine because I am living it xd

[@RayanLaRoze]

Excellent video as always. Just a curious question which software do you use to create these videos ?

[@TheRealXartaX]

Simple derivative/integral calculation/understanding is extremely easy... Just think of it like this: The derivative is the acceleration of whatever function you're looking at. How fast it changes over time (or length in terms of the graph). Once I put it this way I coasted through calculus lol. And the integral is just the opposite. There's no need for a lengthy explanation or fancy visuals.

[@jannikthorsen3531]

This is a brilliant visual tool for learning.

[@hjaiswal768]

I'm always amazed by your thorough understanding about maths...Good work

[@christopherramsey7027]

But if you use the inverse of the operation 1+1/x, which is 1/(x-1), then you'll eventually approach the negative value.

[@rigille]

It's like -1/φ is where all the numbers come from, and φ is where all the numbers are going to.

[@sergiu7120]

the inverse is 1/(x-1)

[@christopherramsey7027]

sergiu coseru thx

[@prabhatp654]

Your prologue made my day!!

[@KaranLobana]

this is a great way of viewing projection as a way of reducing dimensionality

[@anshulagrawal2319]

best video till date i have seen

[@schulzcbs]

What you just explained is called "parallel coordinates" and for some reason US (visualization) researchers strongly dislike them. They are more popular in Europe though and can be super useful (I have a few publications that use them to, e.g., show correlation, clusters-flow in high-dimensional spaces, etc. :) ).

[@Callie_Cosmo]

I feel like this “stableness” of phi and “unstableness” of phi’s little brother closely correlated with that one object that was 3d printed to only have one stable point, and one unstable point, I have no idea how that would correlate mathematically with this but I just feel like it fits for some reason

[@cmyk8964]

You mean the gömböc?

[@Callie_Cosmo]

@@cmyk8964 yep thanks I would never in a million years be able to remember a word like that on my own lmao

[@cicciobombo7496]

3b1b channel best channel Change my mind Pro tip: you can't

[@minotaur470]

I watched this two years ago before calc one and I'm going into dif eq in a month and I intend to watch it when I'm done with linear algebra too

[@sl1pz369]

My Calculus AB teacher made me rediscover this channel

[@asapannarakesh938]

You are my inspiration! Biig hugs from India!!

[@St0n3dCold]

During watching this my brain cells exploded and new ones generated, thanks.

[@201950201950]

Wow I love the graphs. I like to imagine that I actually understand the calculations. Maybe someday.

[@TheAIEpiphany]

Amazing series Grant! We shall remain forever indebted!

[@LordMichaelRahl]

Yeah, you're saying 'φ' correctly and 'π' is also pronounced "pee". I'm sorry guys.

[@angelmendez-rivera351]

Plank Skills *In English it's pronounced like pie* That idea makes no sense because it's not technically a word of the English nor a symbol of the English. The only valid pronunciation is the Greek pronunciation.

[@2k5dcr]

"pee" gets people confused with the difference between π and P

[@streight4lk]

That is true I guess for english speakers if π and p might be used as symbols in the same problem then it is helpful to pronounce them differently. there is no such risk of confusion of φ with any Roman letter so I say pronouncing it either way (phee/phy) is unambiguous and therefore okay

[@daydreamer266]

Your videos gives me goosebumps

[@storskegg]

2:52 - you just explained how a traveling wave tube amplifier works, where those points are electrons bunched as a function of the incoming waveform.

[@isavenewspapers8890]

0:50 "But as you generalize calculus beyond functions whose inputs and outputs are simply numbers, it's not always possible to graph the function that you're analyzing." I think the animation is meant to demonstrate one of these ungraphable functions, but it shows a function whose inputs and outputs are simply numbers-complex numbers, to be precise. Maybe you were specifically talking about real numbers.

[@Nylspider]

OGs remember that the original name for this video was "what they don't teach you in calculus"

[@archentity]

This video helped me invent the world's first warp engine.

[@meghajoshi6442]

Thinking about big brother n small brother, I was vaguely reminded of stable n unstable equilibrium . Strange. Very very nice vdeo

[@simply42]

10:37 The beauty of math summed up in one image 😮

[@HHHHHH-kj1dg]

No not exactly. Beauty of math is not always visual.Its the concepts and relations and mysteries that make math beautiful. Not just a fancy picture. If you think that the beauty of mathematics is summed up in that picture,then you haven't been thinking actively and long enough in math.

[@simply42]

@@HHHHHH-kj1dg Of course that's not everything math has to offer lol. I was referring to math being really beautiful because so many unexpected, interesting things crop up where you'd least expect them. Then, you can understand why these things appear, and even though it makes perfect sense why it's there, it still feels surprising and wonderful. That image was just a really good example of it, the appearance of that elipse completely took me by surprise.

[@markgraham2312]

The technique at 9:50 isn't awkward, it's actually makes a lot of sense. No, the explanation at 10:30 does not give a really neat, emerging pattern. What you mean by "contraction" is convergence. You are creating a brand new term, where the old, identical term is sufficient! At 13:05 the expression on the left, isn't even close to the expression on the right. [Be honest.]

[@ThiagoTabosa]

Fantastic video! Congratulations for put a portuguese subtitle ! 👏👏👏👏

[@clingyking2774]

I prefer this approach

[@dropkickedmurphy6463]

Shit, youre right. Calculus 101 was very rough

[@kummer45]

This is beautiful because it's intended to describe the world of a number we all want to know more. Fibonacci and Lucas Number and The Golden Section Steven Vajda

[@gdgaur]

After 1 min onwards I realised this cant be video which except a math student would love to watch.

[@Hypercube9]

I've been out of school for 2 decades and I still haven't used calculus for anything!

[@skoolscribe3918]

How does the Fourier series explain how hair twist could be grown naturally?

[@Hello33441]

mind blown.....

[@BluespotKneeClinic]

Dear Grant, I am a total novice. I couldn’t help notice the apparent similarity between the outline of the circle/ellipse in this video and that in Richard Feynman’s lecture on why the planets have elliptical orbits. Please let me know if there is really a mathematical reason for this. Kind regards, Shameem Sampath

[@zeropikaify]

14:54 어떤 인간이 저렇게 자막달았엌ㅋㅋㅋㅋㅋㅋ

[@minjeon3037]

한국인을 찾았다!!! 드디어!!

[@zeropikaify]

@@minjeon3037 와!

[@alejandroojeda1572]

Where was this video when I studied numeric calculus?!!!

[@RobertTempleton64]

Have you considered that 'spider web process' as being an attractor in a phase space - although not related to non-linear dynamical system? I may be reading to much into the similarity with Chaos Theory and its use thereof.

[@iloveamerica1966]

10:40-10:53 I see an oval or ellipse and it makes me wonder if f(x)=1+1/x transmutes in some way (by an angle) to the equation for an ellipse f(x) = x2/a2+y2/b2.

[@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..

[@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?

[@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

[@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...

[@AndrewGazz]

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

[@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

[@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.

[@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

[@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..."

[@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

[@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

[@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

[@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

[@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"

[@BritishBeachcomber]

Introduction to Calculus is probably the course everyone dreads. Because it's taught wrong, with little reference to the real world.

[@shivraj6664]

boi gotta flex those conceptual muscles

[@shmendusel]

weird flex but okay

[@syamantakbharti3463]

@Diego Marra XD

[@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?

[@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.

[@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.

[@Brandon-dj9cq]

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

[@rayyan21d]

Dang that's me about 3 years ago

[@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.

[@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)

[@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.

[@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

[@schizoframia4874]

Why is this video so fast?

[@schizoframia4874]

What do you mean

[@govindasharman425]

Wait....who taught him this?

[@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.

[@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

[@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.

[@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.

[@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!

[@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

[@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!

[@LeonKerensky]

"Picture yourself as an early calculus student" yeaahhhh, about that.

[@nitroptics]

Bro why am I watching this I don't even know what calculus is 😂😅

[@bonifaceawa]

If you watch, you might know what calculus is. Doesn't tell you all the techniques and rules for actually calculating these functions, but it is easier to know what is means to know only what it means and the figure out [or look up] rules and techniques, than to memorize all the mechanics, and then figure out what it means.

[@User-ot6jd]

Calculus is basically cutting up things into really small pieces (differentiation) and also adding up many small pieces into a large piece(integration),pls correct me if I'm wrong.

[@PaulPaulPaulson]

Things like this have a lot of applications in control engineering.

[@yousafzai8547]

Paul Paulson really? im interested. can you provide an example or a good read? #physicist

[@JSnacko]

Adnan Hafeez One example can be reactor temperature. In a lot of chemical reactors there can be multiple temperatures with steady state operation. Some of those points however are unstable while other ones are stable, each temperature giving you different chemical conversions. So trying to control the temperature at an unstable point is very hard due to a slight deviation resulting in a new stable temperature that could be drastically lower or higher than what you initially had. If that's not what you were looking for or what the OP was at all referring to them sorry for the wasted comment

[@PersonaRandomNumbers]

Oh, definitely. Immediately as I saw the part at 8:15, I started thinking about control theory and stable vs. unstable equilibria. A pretty standard example of control theory is a pendulum. A pendulum has two independent dimensions -- angle and momentum. There are two equilibrium points -- resting at the bottom of its swing, and standing perfectly straight upright. A small pertubation near the bottom will leave the pendulum still resting at the bottom -- but a tiny nudge at the top, even from something as small as thermal noise, will quickly send it crashing down. The difference is huge, and has to be appreciated for control systems, unless you want it to spiral out of control. Adding a controller changes the system, and can add or remove equilibria, which may or may not be stable. The general goal is to remove all instability, but without slowing down the response significantly. Here's a video showing a controller applied to an inverted pendulum to make the system stable: watch?v=855O9x0Pgf0 And a wikibooks textbook with a good introduction to the topic: en.wikibooks.org/wiki/Control_Systems

[@oldcowbb]

poles and zeros intensified

[@robintaylor3713]

after brexit control engineers will only have to worry about the zeros

[@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.

[@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.

[@jorge_cazares]

I'm still waiting for the video explaining the Jacobian :(

[@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.