The paradox of the derivative | Chapter 2, Essence of calculus
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- เผยแพร่เมื่อ 28 เม.ย. 2017
- What is an "instantaneous rate of change" when change happens across time?
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Note, to illustrate my point for the target audience of a new calculus student, I discussed a hypothetical speedometer that makes distance measurements over a very small time. Interestingly, most actual speedometers in modern cars work by analyzing the induced current of a spinning magnet, which is in some sense the universe implementing the derivative.
Thanks to these viewers for their contributions to translations
Dutch: @LFWarsen
Hebrew: Omer Tuchfeld
Italian: mulstato
Vietnamese: @ngvutuan2811
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I'm genuinely jealous of the next generation of students who will have these videos to introduce them to calculus. The understanding and intuition they bring to a newcomer must be really exciting.
Devin Neal It is!
I was first exposed to Linear Algebra by his video series. It really got me hooked and I looked forward to every video curious about how the subject will evolve. That was really awesome! I bought some books on mathematics and self-studied maths quite a lot since then.
I'm going into a-level further mathematics next year after GCSE and I am so grateful that this guy does what he does so well
They're exciting for me as well, and I really enjoyed calculus back in high school.
Devin Neal they're very exciting
That's it, I'm promoting this channel wherever I find people who say " I'm really not a math person"
I had the same idea but some people are really not into it.
I was 'not a math person' in high school because I had a crappy understanding of algebra and equations, but always excelled at the theory stuff regardless. I eat videos like this up because you learn all the theory and abstractions that show how cool math can be, but someone else crunches the numbers for you. They should really use this as learning material in schools, it could help a lot of students like me.
Josh Tipton as a person like you who loves theory and abstraction but not so good at number crunching but is about to be put into the meat grinder of high school math I cannot say how much I am glad to be at this point of history, a point at which pure math and implied math begin to cross along with so many people and resources to help me to grow my understanding. If you have not then I recommend this video - th-cam.com/video/OmJ-4B-mS-Y/w-d-xo.html
I'm really not a math person and I don't understand it ;_;
I think you will be surprised at how much someone would still dig in their heels. Learned helplessness is a very deep hole made be habit of thought and preconceptions of their ability and of the subject matter.
When I watched this years and years ago I fell in love with maths again, now I'm a maths teacher and I'm passing this down to my students. I'm so grateful for your channel and I hope you understand these are more than just TH-cam videos, they mean a lot to people.
That's incredibly touching to hear, thanks for sharing!
@@3blue1brown respect for replying after 6 years since the post date
Just imagine being 3Blue1Brown and reading such comments from time to time. Just imagine.
@@3blue1brown Grant, It was just wow ❤❤... Since you've got this exceptional talent of explaining and animating So why not try some topics like The Essence of complex no....
0: Oh? You're approaching me?
dt: I can't beat the shit out of you without getting closer
0 : Oh ho, then come as close as you like...
I love this community. Making jojokes on maths xD
JoJo references on maths related topics is something i expected when certain zeppeli started using the golden ration on us mortals, then mr Jonathan started flexing his spin.
This was hilarious
Oh shid I never thought my JoJo fascination was gonna follow me here in Calculus 🤣
"I would use the letter "d" for distance, but that guy has another full time job in Calculus" Lmao
I was thinking I was the only one who would laugh at that.
Isn`t "d" a minor? Isn`t it illegal to make children work full time?
@@teeraxgaming Isn`t "d" "D`s" son?
@@teeraxgaming Save the children!
@@teeraxgaming
Why??? Are you kidding? Because
1. They give young parents extra time at night.
2. They make middle aged parents go back to calculus!
3. They keep mature parents from buying a Porsche or a villa in Italy and invest their money in college tuition.
4. They help old parents formulating their last will.
5. And they keep pediatricians happy.
That`s why!
Grant, I don't think I've ever watched a 20 minute math video and wished there was more. You are super talented, and I want to thank you for releasing this to the world for free.
This is not just math. This is art.
ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟ ᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟᅟ He is Dr. Art
Math is art, and he's an artist
His videos are just so awesome that I'm gonna cry
I want to like this but you have a perfect number of likes and I’d rather not break it
Math is art. And art is math.
watching this series has made me fundamentally question the quality of the education I received at school growing up. It's exposed how badly we were explained these concepts in class. I did so much calculus at school without ever being given a real understanding of what i was doing. And then I wonder if the same maybe applies to all my subjects at school. maybe my whole education was actually entirely sub par and I've fallen far short of my potential as a result lol.
Welcome to humanity!
@Mr Right yeah thats very right mr right
@Mr Right Also, watching this, people are relaxed, can pause and replay etc.
@Mr Right It should be possible for 12 year-olds who think they have all the answers to go out and get a job, an apartment, etc. If they want to continue their education later as an adult, it is still available and fully paid like grade school was.
Your teachers were mediocre and apathetic
Thank you for the great quality videos. It makes things easy to understand.
Fr
ong
for a second i thought that it said $1000
@@sandhan1t1zer But it's worth more than a thousand.
@@johnchen2325 sand hanitizer?
Grant, I'm a 39 yo man in the verge of changing my carrier from the Ad industry to the Dev and AI realm. I really wanna thank you for your overwhelmingly great work. This video, oddly enough, changed my life, and I'll forever remember you as a positive force for this change. Thank you very much.
Ad AI.wow
You could almost say that after this video the derivative of your feelings with respect to time is positive
I am 53 and I am just relearning maths as a hobby. I'm doing it for 3D art and animation including physics simulations that require maths.
All the best
man this is fucking powerful
Those "ohhhhhhh, aha!" *click* moments that were so rare in my University calculus classes are so frequent in these videos.
University? I thought you learn this in 3rd grade of highschool. Im in 8th grade rn. Do I srsly have to wait 5 more years for this?
@@methatis3013 Not everyone has American style of education system, Calculus is elective in my country for example
@@methatis3013 i study this in 11th class in india .
PrimalForlorn In the UK, calculus is introduced in sixth form (year 12), but of course it carries on to university level
WIsh I had all these TH-cam Videos when I was still at school like 10-5 years ago.
I loved math and physics.
(I studied math and physics at the university afterwards)
The way those subjects were taught at school was plain and super boring.
All these videos on TH-cam are so fun and I learn even now from all these.
I believe in the next decade or so - some really clever minds will arise - smarter than Einstein and Stephen Hawking and all others.
Just due to the fact the way all the knowledge is provided and super easy to learn and visualize with the current age of technology.
It literally wasnt like that even just a few years ago.
The future is looking bright.
everyone: "no one can calculate instantaneous change. it's totally nonsensical"
fathers of calculus: "so I'm gonna do what's called a Pro Gamer Move-"
The art of problem solving is solved here, no problem to show !
Wrong. Makes sense so well as work at derivative. See My 3 years ago post, also i posted at keystone video about paradox of derivative.
*Pro Mathermatician move
@@Brahvim *Pro Mathematician Move
This comment is now 2 months old, thanks for the correction, @@user-yj8uv7gi3o 😂👍
(Saw this after 3 hours :|)
Wow I've almost obtained my bachelors degree in Aerospace Engineering and I finally get to see why the derivative of a polynomial is the way it is, thanks for your great videos and insights!!
This gives me hope hahaha
@@theguybroseph thats great little Ben
Can you give updates on the searched/found jobs once you've got your diploma, please.
Embarassing.
Thank god ,we're all safe to fly in your airplanes now💀
I'm not sure which one I find more amazing: the math itself, how clearly you explain it, or work you put into the animation.
Both
WIsh I had all these TH-cam Videos when I was still at school like 10-5 years ago.
I loved math and physics.
(I studied math and physics at the university afterwards)
The way those subjects were taught at school was plain and super boring.
All these videos on TH-cam are so fun and I learn even now from all these.
I believe in the next decade or so - some really clever minds will arise - smarter than Einstein and Stephen Hawking and all others.
Just due to the fact the way all the knowledge is provided and super easy to learn and visualize with the current age of technology.
It literally wasnt like that even just a few years ago.
The future is looking bright.
It's all amazingly awesome !
3Blue1Brown, I'm a master's degree student in theoretical physics with 1 year left, and probably not in your intended target group. Despite many of your videos covering "basic" (read: essential) topics, I find them fantastic tools for refreshing my knowledge, and even learning something new. For example, from your series on linear algebra, I learnt to visualise the columns of matrices as transformations of the respective unit basis vectors, which has been a great tool for when I have been reading group theory and quantum field theory, even though it was such a simple observation. Thank you for these videos! I look forward to more!
Well, I would consider myself a quite good math student and yet I can still learn much from these videos despite "knowing" most of the content already.
+Elchi King Yeah, even though you already know through theory what the formula of the derivative and such things is, having such a great visualisation of it, really makes things easier when trying to learn and picture new you learn related to that topic.
Imagine that you yourself are a graduate student in theoretical physics. You want to express your appreciation for a video, but you want to actually frame it in the context of why you appreciate it. Just saying "cool video dude" doesn't really do much. Taking the time to explain why something is useful despite not being a member of the intended target audience is far more useful to the man who makes these videos, and it is undoubtedly nice for 3blue1brown to know that his work is appreciated by people in academia as well as high school students cramming for their AP test. If someone pursuing a degree in theoretical physics cannot even state what they are doing with their life without apparently sounding like they are bragging, then it is truly a sad state our society is in.
I agree with Kyle Poe. There are plenty of people out there who, beyond being good math students, live and breathe this stuff every day as part of their profession, and really do pretty much know everything in these videos already at a deep level. After all, the author can't be the only one in the world qualified to make a video like this, can he? But these videos are still so well constructed and explained that they're worth watching for a fresh take.
In one of the linear algebra videos when he said eigenvectors stay on their same span during a change of basis which means that they're just the axis of rotation, I think my jaw literally dropped
I'm studying engineering 10+ years after leaving school. This series is a blessing! I never took calculus or advanced maths fomally and I could not properly grasp the concepts through the book material. Seeing this laid out in such a clear, visual and logical way is sublime. Thank you for taking the time to create such in depth videos.
I definitely agree with you
I took calculus in college. The professor explained how to find the derivative of a polynomial. I didn't learn until many years later, after I graduated, what a derivative actually is.
Then you just didn't pay attention or read the book.
It literally took me months to understand this video but I’m super glad I do now. Thank you so much 3blue1brown
I have a bachelors degree in mathematics, and even though I was a straight A student, I never really "got" why we did what we did to solve problems, just that we had to do it.. until your videos.
Thank you for giving me such clarity on a subject I love with all my heart and a full understanding of why problems are solved the way they are!
You rock.
I started watching these videos when I started learning calculus but I stopped because it conflicted with the way I was being taught. It's a better experience after having gone deeper into calculus and watching these again. I'm solid enough in it that I can keep track of the different mindsets.
@@cubicinfinity You are absolutely right bro. Me too had the same experience. In fact watching these after all that taught in our schools/colleges is really making us fall in love and admire these concepts
@@carlgauss1702 How did you get a job then?
Well....I think you suck.
@@sidharths9416 my teacher just gives us a bunch of formulas lol
59 years old and Grant is taking me to school. Thank you!!!
Every student who says "Math is the toughest subject" really means that their Maths teacher did NOT do a good job teaching them math. Am one of those students! This video rekindled my interest in Math as an adult. I wish this existed during my school days
Or the ones that didn't practice enough...
sincerely,
A student of the latter kind
The people with such deep knowledge will ask heavy salaries. Bcoz they know their worth. And schools cant afford such teachers. Which is why we get poor education from teachers with poor
Bullshit. Math IS the hardest subject, that's not the teacher's fault, but simply in the nature of the thing.
You won't find a mathematician, who is not vastly above average intelligence and thinking about problems all the time.
@@shadymello9146Its not only about practice. If a student doesn’t understand irrespective of practice for so many times & if a teacher isn’t good enough to make the student understand, the its not the fault of that student.
No...most people are just really dumb when it comes to math lol.
"The Essence of Combinatorics" is something I would really really love to see from you!
Keep up the good work!
Yes please please
Yup
notice the point and draw a line.... time is created from 0 to 1 too... i dont remember a nicer conciser version of "how to press start" dug deep, dude
Yes yes yes please. Please cover vombinatorics and probability.
Fact: Most adults could never pass a course on Intermediate Combinatorics
I think it's safe to say that by impacting millions of people watching your videos, you've single handedly made the world a little bit better place.
I'm 14 and love these videos-they really help me understand calculus though especially since this is a series I would say a bit more pausing and giving the viewer a chance to work it out themselves would be helpful to keep engagement, but apart from that these videos are amazing and I love this series, even six years after you released it! Thank you so much!
omg this channel makes me think how poorly these concepts were explained in school. Keep up the good work.
The people with such deep knowledge will ask heavy salaries. Bcoz they know their worth. And schools cant afford such teachers. Which is why we get poor education from teachers with poor knowledge.
Nice video.
My University's teacher used this to intoduce us to derivatives
WIsh I had all these TH-cam Videos when I was still at school like 10-5 years ago.
I loved math and physics.
(I studied math and physics at the university afterwards)
The way those subjects were taught at school was plain and super boring.
All these videos on TH-cam are so fun and I learn even now from all these.
I believe in the next decade or so - some really clever minds will arise - smarter than Einstein and Stephen Hawking and all others.
Just due to the fact the way all the knowledge is provided and super easy to learn and visualize with the current age of technology.
It literally wasnt like that even just a few years ago.
The future is looking bright.
If I only had the same teacher.
He/she should get a reward.
He/she should get a reward.
uhh, me tooo ahaha
This reminds me my high school teacher who was really great at teaching, instead of force feeding us formulas to remember, she would first show the proof of why it is what it is. The first time I saw a mathematical proof it was mind blowing.
Unfortunately not many math teachers actually even know proofs, and most kids around the world are getting gavaged with rote memorization of terminologies and formulas they don't understand... :(
tbh not everyone needs the proof, alot of us just studied maths to pass and we really couldnt be bothered to learn the history and context
@@evanchong6482 that is exactly the point. "Studying to pass" makes no sense. It is the same as saying "living to die". Study should be about proof, about curiosity, about exploration. This is what makes humans a miracle and not slaves of some nonsense way of living...
As changing my career from a Data Analyst to a Data Scientist, this video really helps me to brush up my calculus to learn advanced statistic models. It changes my future, thank you for interesting and awesome video !
I love that even though I've been an avid student of math for most of my life, preaching its wonderful usefulness to anyone who will listen, your videos will invariably contain 50% of things I have learned, and the other 50% wisdom that is gained only by tying all fields of math together to produce a bird's eye view of a master concept. That extra amount on top of the base material represents you trying as eloquently as is humanly possible to share your bird's eye view with the world. We are fortunate to have your passion and knowledge used in this way, and we can only say "thank you!"
I know it’s a bit late, but may I just say that this is an absolutely phenomenal series. It really helped me understand calculus in a way that just didn’t happen at school.
At school they teach u for exams that's why.
In college my Calc 1 professor led us to the point where we could see the derivative for ourselves. When, in a homework assignment, I did just that, he implied that I must have taken calc in high school and been pretending to have had that breakthrough for myself. A real kick in the jibblets. Some people just shouldn't teach. When I read your intent for this video series, it really hit home! Thanks for allowing folks to experience the wonder of discovery.
THE 👏🏻 FACT 👏🏻 THAT 👏🏻 THESE 👏🏻 VIDEOS 👏🏻 HAVE 👏🏻 NO 👏🏻 ADS👏🏻
Wow I thought I was the only who thought the term "Instantaneous rate of change" didn't make much sense.
Instantaneous slope at this point of curve !
Instantaneous velocity exists and make sense. I posted why in the keystone video about paradox derivative, as here 3 years ago.
Here I am, Professor, taking notes as if my life depended on it...Thank you very much for an excellent, much expected and very illuminating series!
I “learned” derivatives at school today. My teacher said to write down the definition on the board but not the crossed out portion which read “it’s almost like finding the slope at a single point.” And then he gave us this formula, f’(x)=lim(h->0) (f(x+h)-f(x))/h. I spent the example problem time as time to figure out how this equation gets the derivative.
Thank you for showing what derivatives are and not a random equation that means nothing to most people.
At least you remember it. I am sure we had a 'slope of line on curve' lesson and think "um ok why, just playing around with tangents now?".
Well its only the foundation of classical physics, involves infinite limits and all sorts, but anyway...if only i knew at the time instead of "No JuSt LeArN iT"
I am 13 years old and I Would like to say thank you for this incredible series it has helped me a lot in my Studies of mathematics
Mf is 13 and doing calculus
ok
keep it up!
beautiful! lots more cool stuff to unlock :)
He makes magic using just a bunch of living pi symbols with eyes and some really good graphical animation. Hats off. Never going to forget how you helped my in my quest of ultimate knowledge. Thank you. Makes me question what schools even do.
Relatable with me. I wanted to learn calculus because it just passed through my mind one day. I thought I would learn this after a some years and that learning it is impossible. But as I begin learning, I realised that this branch is the most beautiful branch in all of mathematics. And yeah, I learned this for the purpose to use Integrals in my work and for ultimate knowledge.
Being an undergrad student, studying at a level a little more advanced than this, your videos really make me take a step back and appreciate the beauty, a sort of 'stop and smell the roses' for math. Thank you, 3Blue1Brown for being a brilliant educator.
Same here. I first watched this video when it was published almost exactly two years ago, and coming back to it now is something really beautiful. At the time, whilst I could follow it and gain a lot of intuition, it seemed quite complicated and it was difficult to really see how it fit into the ‘bigger picture’ of mathematics. Now I’ve nearly finished my first year of undergraduate mathematics and been introduced to the more general principles and some of the more rigorous constructions of calculus, this is just wonderful to see again. It’s encouraging to see how much progress I’ve made personally in my understanding but also to see how almost exactly the same techniques used in this series can be applied in so many ways to obtain all sorts of other incredibly beautiful results in other areas. The fact that much of mathematics is so ‘useful in the real world’ seems like a wonderful coincidence, because even if it wasn’t... it’s just so....... lovely. ♡
I'm a french engineering student, and despite the fact that I already know these things, I love watching your videos because your way of talking about maths and illustrating them really makes me wonder deeper questions than what I already learnt. Great job I love your channel 🙃
Evi1M4chine This probably sounded a lot more passive-aggressive in my mind than I think you intended it to? I would say that engineering students do understand things deeply, but there will always be people, in any field, that simply accept rules and move on. But again, that's only a few people, not all people. Videos from this channel are made for the purpose of curbing that, however, and that's lovely! :)
Paragraphs please. My eyes hurt.
E: On the other hand we've got to ask this: Should education teach them how to live, or should it teach them how to think?
Evi1M4chine I 100% agree with Yue Chi K, I think in any field there will always be people trying to cheat as you said it properly, but they won't be good engineer, surgeon, teacher, etc...
According to me, some professions require passion or at least interest to be able to work in correctely, and engineer is one of them, I hope your flatmate won't realise it too late.
+
This is brilliant - I have struggled with calculus and at the age of 61, this has made it so clear! Today's students have some tremendous resources available to them.
This is great! I used to study maths at school in such a bad way, through memorization and solving problems in an "automatic" way, without studying the actual reason and meaning of mathematics. Now I'm at university, reviewing again some fundamental topics and I'm seeing maths in a completely different way. Your videos are great and are helping me a lot to discover new things. Keep up the great work!
Ps: Sorry for some grammar mistakes, I'm learning English.
We are truly honoured and privileged to have access to the thought processes of the giants who have gone before.
"Sometimes when everyone is standing on shoulders of giants, it is better to be an ant." - ThinkTank255
What this means is that it is great to make progress based on the works of others, but sometimes, we must challenge and question even the giants to make progress. Remember, these giants were humans, just like you and I. The idea that we cannot improve on the foundations of mathematics itself should never be assumed simply out of honor or veneration of the great men that came before us. Indeed, I think many of them would themselves be honored to be challenged and even proven wrong.
I think mathematics has, to some extent, lost its way. It has been standing on the shoulders of giants for so long, nobody knows how to be an ant. There is a lack of critical thinking, even within mathematics itself, which is actually quite frightening. To put it metaphorically, standing on the shoulders of giants leads one to believe that maybe the giants shoulders are the only way to elevate oneself and maybe no mathematics exists where giants don't exist. In contrast, I think mathematics exists everywhere, and we can invent new mathematics and new ways of doing things if we try hard enough. We need not be bound to the dogmatism that exists in mathematics.
Well actually... Human brains evolve over time... Meaning most of us are smarter than newton and the others.... But we are a light year behind their determination... Many of us are just too lazy and thats true and its a fact
@@ThinkTank255
It's not convenient for a person to do this though. You're right though there may be other crazy math that we will never find.
I like the black background. Bc i see how stupid my face looks when I'm watching those kind of videos 😅
It's a whole army of stupids then mate, onward my fellow viewers!
lmao
This is why I dislike the black background
lol
Best comment
I have no doubt that this is one of the most clearest and satisfying fundamental math topics videos available in the planet. What an amazing useful and enriched piece of content. Thank you so much Grant, now I have totally understood this concept on a intuitive way
Thank you so much for this. Your explanation makes so much more sense than "decrement by one and multiply by the old power". It's far better than just regurgitating what teachers say.
10 days of Calculus videos, 10 days till the AP Calculus Exam. Coincidence? I think not!
lmao, I'm just waiting for him to explain deeper Euler and Newton's method, and the Lagrangian Error Bound.
newston's method is not an essence of calculus
Erik Beserra no
yo soy arból 5 lol is the second-worst grade here
Erik Beserra .
Next up with be "Derivative formulas through geometry". Keep track of the full playlist at 3b1b.co/calculus
If you are curious, as some commenters have pointed out the way an actual car's speedometer works is to induce a current with a magnet whose rotation is determined by the rotation of the wheels. Depending on your view of the underlying reality behind certain physical laws, some might call this a genuine "instantaneous rate of change", but it still leaves the question of what exactly that phrase means.
As far as setting the stage for learning calculus, and inventing the math needed to capture this concept, I do think it's important to recognize that the idea of a "change" makes no sense when you blind yourself to all but an instant. This is why the derivative of a function, which extends the idea of a rate of change to make sense for a single instant, is intimately tied to the function's value around that point. Perhaps there is a statement to be made about how physical laws determining the state of a system at a given time are intimately tied with the surrounding points in time, but that is beyond the scope of the series.
great video
There is a mistake in the notation at 13:30, shouldnt it be ds/dt of s(t) not ds/dt t. or d/dt (s) or d/dt of t^3
At, 9:13, how come you calculated the derivative at t=10, isnt the graph discontinous?, also, by your intuition(which is best), we can't extend ds "just" above t=10.
BTW great video love ur stuffs!!
The graph isn't discontinous at t=10, we just looked the function in the intervall (0,10) and it isn't even clear wether 10 is in the intervall or not. You can define the derivative even if the intervall is closed, in which case you just have the limit from one side in the boundary points
dt is not a number! Thus it makes no sense to say its zero, and also no sense to subtract soemthing from it. It's just notation for the limit
0:00 intro
1:07 central example
2:20 velocity
3:26 change in time
4:32 ds/dt in the real world
7:13 tackling the paradox in pure math
9:44 the true derivative
13:11 take a step back
14:18 "instantaneous rate of change"
16:23 outtro
These videos are super good, this feels like something that I would have to pay for on an online course. I originally had no interest in calculus, but now I love it since you're always able to explain it with real world examples that are easy to visualize and understand. These make me really look forward to when I take calculus next year!
OH MAN, THIS IS GONNA BE GOOD
Ethan Rojek You be right mate!
HELL YEAH! LET'S DO SOME CALCULUS!
Ethan Rojek I CAN'T WAIT UNTIL I BEGIN SOLVING THE 3x+1 PROBLEM USING CALCULUS!!
Evan Bialo AWW FUCK YEAH CALCULUS
kkk
Your work here has permanently changed the way I think and how I feel about Mathematics and life in general. Please keep up the good work. When you feel down, please remember that there are so many of us that are grateful of what you have done here.
Man, I am so grateful and thankful for your explanations. I've never felt enjoyment of learning something in last 4-5 years like this.
Math has never been this exciting to me !!!
I'm so grateful that channel like this does exist on youtube
Love from Vietnam
"when you bind yourself to all but just a single instant, then there's not really any room for change."
true in math and in politics.
Sameen Dusk We live in a society.
Are those any different?
hMmmmmMMmm
ew politics
@@MrOdsplut we live in a society
Thank you so much for calling out the "instantaneous rate of change" oxymoron. I pointed it out in to someone and they overruled my point simply because they pass exams better than me
I've almost burst into tears during watching this. The derivatives have never been so clear to me. Thank you so much, exactly what I've been looking for!
since is so clear for you maybe you can help me .sorry for my stupid question .in this video the derivate is 12 ..but 12 what meter for second ?it should be the istantaneous speed and so how is possible it s the same at any point?thanks in advance
@@mauriziop4307The slope of the tangent line to the graph is 12, he just ignored all the terms with dt in them because dt is approaching 0, so it's a REALLY small number, but not 0 and not infinitely small too. So those were safe to ignore
Damn this is amazing. I have a calc test in a few days and I don't wanna rush any of this. It's to beautiful and I watch each episode like 5 times to really grasp the idea. I usually shrug at patreon but once i get my pay check i have to support. Thank you so much for helping me understand, but mainly, helping me appreciate math so much more.
Your animation bringing the secant line down to a tangent line really drove it home. Thank you so much.
LOL physics students be mad
speed and velocity thrown around everywhere
I actually enjoy velocity problems lol
And the use of distance and not displacement, but this is an amazing video (and channel) lots of love!!!
yas
@@katienewman4743 them vectors are a bitch.
speed is the magnitude and velocity is the direction with magnitude.
He has truly achieved his initial goal of the series. I do feel what those mathematicians would have felt just after discovering derivatives and the unique formulas for a few of them. Congratulations. Hats off.
"It´s kind of FLIRTING WITH THE paradox of change in an instant whithout ever needing to actually TOUCH IT " 3Blue1Brown
That was funny :D
The choice of colours for “instantaneous” and “change” was really clever. I love your attention to detail. Thank you so much!
Anyone can be intelligent, but only those who can successfully articulate their incredibly intelligent ideas are brilliant. Great video 👏
ese est percepì
"If you can't clearly, and succinctly, excplain a concept to someone else, then you don't truly understand it yourself." - Paraphrasing 'someone'.
@@AthAthanasius Thats exactly what i wanted to write
@@AthAthanasius Richard Feynman
@Emma Wood .. Do you mind if I steal this quote for my book, desktop, discussions.. and pretty much everything?
I love this series! I am currently taking AP calc in high school, and this is the most sensical explanation of derivatives i've ever seen!
I just would like to say thank you so much, I have watched these over and over and each time I feel like I understand a little better... It makes it so much easier when you REALLY understand it.
One thing...Please don't stop making this kind of videos🙏
Being a mathematician working on the financial industry (and despite what you might have heard about financial markets using maths quite often and rigorously) who wants to recover part of my former mathematical reasoning and intuitions lost among nonsense, nonrigorous and nonscientific economic pseudo reasoning, I couldn't have stumbled upon a better resource than this channel. I will only say congratulations, because no matter which words I pick, they will do no justice to the art performed here. As announced, congrats!
This is incredibly good teaching. Even after 4 years of an engineering degree I still found myself relying on the higher-level rules of calculus without paying enough attention to where the rules came from.
This has finally sewn together my learning in a way that will stick in my mind. Thank you.
It is astounding how much better of a teacher you are than my calc professor. Thank you for taking the time to make these videos, the level of intuition I have received from this series has been incredibly helpful. Maths are taught like shit in universities - I know this because watching your videos for an hour has given me a more in depth understanding of these concepts than my entire semester of calc 1. You sir are a legend. Anyone who is struggling with calculus should stop what they are doing and watch this series in full.
When I learned calculus, for t>>0 instead of d(t) we used Δ(t). This notation was helpful for me to understand the concepts you discuss so well.
"Sleeve of Wizzard" - Borat
My calculus teacher teaches us that the reason we use "d" intstead of (delta) is because d is latin script and delta is greek script... don't know if thats helpful
@@vincentstone7272 Except in mathematics, except for the conventional ones (e, π, special functions, etc.) you could use any of the Greek scripts or Latin scripts as symbols for mathematical objects. It's perfectly valid to write τ^2-τ-1=0 or "let Λ be a connected open set" or "Θ(x,y)=χ(x)ζ(y)".
@@vincentstone7272 @nomi udo
Δ(t), represents "a defined change in t", whereas d(t) represents "a change in t that approaches 0". Grant used d(t) as if it was Δ(t) in this video for informational purposes, but really saying that d(t) ---> 0, is redundant because that is what the "d" represents. Whereas the "Δ" just represents a change that does not necessarily go to 0 unless you take the limit of it. In other words d(t) is a shorthand notation for the limit as Δ(t) approaches 0.
my mind was literally blown when he explained how you can just... ignore terms containing dt
so intuitive!
I had to stop the video at that moment. Wow
Same!
(I realized how you can always break down a fraction later)
I m in last year of my high school.,.and found this series....it's never too late... :)
Sameeeeee!!!!
I am in love with this method of teaching and the mathematics visualization program you created, Manim, is amazing! I am going back to school to learn Computer Science and I am excited to know that soon, I can begin utilizing Manim as well! Thank you for everything you do.
Fun fact: The reason the displacement function is denoted by s(t) goes back to the German word for distance, "strecke" (despite the fact that this is the word for distance and not displacement).
Are you sure it doesn't come from the Latin Word "spatium"?
As a precal student, your videos actually hype me for calculus next year. I love to be ahead and have a clear understanding of calculus going into school next year.
@@DoesMahBlockLookBig what didi you learn in school until this point, did you get to integrals?
star of duty Just yesterday we touched up on the basic definition of a definite and indefinite integral. Right now we’re taking baby steps by using Riemann Sums and Trapezoidal Rule to help us visualize how integration works.
@@DoesMahBlockLookBig since english isn't my first language, I don't completely understand what you're talking about.
I don't think we even learned what integrals are except for "the opposite of derivatives" or "the area between two graphs"
@@terner1234 Riemann sums are what was discussed in the first video -- an approximation of the area under a graph as a sum of the areas of rectangles, calculated by (a) splitting the part of the graph for which you're measuring the area into several pieces; (b) choosing a point on each piece of the graph; and (c) calculating what the area would be if each piece were replaced by a horizontal line passing through the chosen point. (If I recall correctly, the actual definition of an integral is the value that the Riemann sum approaches as the width of the widest piece approaches zero.)
The trapezoidal rule is a similar idea, but with trapezoids instead of rectangles -- instead of replacing each piece with a horizontal line through a single point on that piece, you replace it with a straight line connecting the endpoints of the piece.
Then you will be that annoying know-it-all student who sits in the front and tries to lecture the professor.
I'm currently taking Calc 3 and I still had questions about what limits and the derivative really meant. Calculus has such an amazing background, such complex implications of what we think about change and infinity, and yet many teachers just swift through it as if it were nothing.
Thanks for doing these videos. Knowing that people are out there working for quality education is such a relief when regular class periods or teacher schedules aren't enough for engaging in specific topics and doubts :)
Skill issue.
This method of explaining calculus really just makes sense, it's so elegant and simple yet still tackles the core elements of what calculus is, it has really helped me learn
The amount of times I went like 'oh' and 'aha' and 'wow' says clearly about how much of an enlightening experience this was. Thank you so much!
Even though this video explains a concept in an amazingly beautiful way I still have to watch it multiple times to even kind of understand what he’s talking about
Absolutely fascinating the way for you to explain these concepts. If I just had been explained this way, I would for sure be a genius creating ultra mechanical devices very helpful to the people. Now is way to late but the fascinations persists. Thank you for this outstanding work.
I am incredibly thankful for this series. It's hard to find teachers that teach at such a level that cares for true understanding
6 minutes in and I've already learned more about rates of change than in my high school and college calculus classes combined
SOOOOOO glad I found your videos. I am a science teacher, and my math has gotten rusty over the years, so I am trying to dust it off improve. You are bringing so much clarity to intuitively difficult concepts. Thank you!
how does this guy teach better than a professor of 23 years...
i learnt now that the concepts i have built by practice in math could have taken a lot less effort and sleepless nights, and i might even have gotten better!
keep up the great work, Grant!
WOW! Fantastic explanation. I've been trying to understand how to calculate derivatives since I took Calculus in business school 30 years ago, and you just clarified it all for me. Wonderful job! Thank you!
I've always wondered whether it's possible to do an absolute perfect and flawless math instructional video. After watching this one, I know it is.
The way you present the information in such an organized and compelling way is incredible
Thank you. I am reaquainting myself with the basics of Calculus in a Calc 1 class. I took courses in high school nearly 2 decades ago and had senioritis the entire year. I love your reasoning and your intuitive problem solving methods. I will recommending your channel for further conceptual development for any of my classmates.
man if there is a heaven I am sure you will go up there. I am forever grateful to you for building this concepts with in me. I think a thank you would be small but from the bottom of my heart thank you, you are great. supporting you on patreon is my one of the bucket list once I get a job.
This is the kind of tutorial that makes me love mathematics. Congratulations for the great video
I've never been so excited for a video series! :o
I have... for the last video series of 3Blue1Brown :)
I am in pre-calc right now, and this is soooo helpful! Thank you
If you're watching this, and you don't already know what calculus exactly is, consider yourself blessed to be introduced to it in such a magnificently crystal clear way.
The music + Grant’s voice + the visuals + the content make this channel my favorite math TH-cam channel.
This whole video is a "d(OMG)/dt" moment!
Hahahaha so true
x=OMG
d(OMG)/dt = 0; in other words, CONSTANT OMG with respect to t :)))
@@claycurry5782 which means it's going to be always OMG ;)
That is great.
Finally someone to explain how things work and where they come from.I am so happy that there are such creators like you because at school no one explains where things come from,and while for someone that might be okey,for me it means to lose my mind while trying to understand how and why things behave like this.However that's how i started to love maths,when you understand something,it makes you feel amazing when solving a problem without using memorized formulas that don't make any sense for you.I wish they taught us better at schools and not make kids hate education
12:22 this was the moment that made it all click. the factoring, and then treating the dt terms as 0 (since the limit of dt is 0). Once laid out, it looked so simply, yet so brilliant. I wish I had seen this video sooner. Thank you 3b1b
Calculus is every maths lover's favorite topic in secondary school because it's just so beautiful...
Balázs Novák & in Electrical Engineering studies :)
+Evi1M4chine thats the genius behind the limit.
wait, we call it zero but it is actually not, in reality it is a very very small number. the whole idea is that the smaller it gets, the closer the expression gets to a certain value what we call the limit of it. this is also mentioned in the video. we pretend it's zero, but it's actually not, but you got the essence of the paradox right :D also, this is the beauty of it, because it works perfectly
Ye, it´s so fun. Yesterday I did 8 hrs straight of calculus exercises. Not even kidding. Not studying for an exam either.
There is no taboo. Division by zero gives an undetermined form that is useless in most contexts and useful in some, such as limits. Also, you never actually have to divide by zero to calculate a derivative, since the denominator cancels out as you can see in the video.
I'm studying for my master's degree in theoretical chemistry and I love watching these videos. The "Essence of linear algebra" helped me to understand change of basis and linear transformations like no other video could.
Thank you so much and keep up the hard work.
same for me tbh
At 1:52 "But that guy already has another full time job in calculus" 😂
These are bar-none the best math videos I have ever seen. Thank you, Grant! You are a national treasure.