Visualizing the chain rule and product rule | Chapter 4, Essence of calculus
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- Опубликовано: 19 июл 2024
- A visual explanation of what the chain rule and product rule are, and why they are true.
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Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced.
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Timestamps:
0:00 - Intro
1:48 - Sum rule
4:13 - Product rule
8:41 - Chain rule
14:36 - Outro
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: adilatte, ang
Vietnamese: ngvutuan2811
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Next up will derivative of exponential functions. See the full playlist at 3b1b.co/calculus
when are you going to start working on real analysis? I'd love to become a part of the community that helps you develop that!
Never change.
I'm very curious how you're going to do that. I wonder if you'll define e with a series and then prove d(e^x)/dx = e^x or if you'll define e to be the number such that this property holds.
will you cover x^x ? :)
I did the time-stamps table for this video too, could you pin it please?
I've been a full time professional math tutor for 27 years. This video series is superb. They will ruin my career, but at least in a very classy way.
Dan Sanford 😅🤔
You can tutor me if you live in Southeast England
It is amazing, but the problem is, I don't think students learning calculus will understand/appreciate it unless they already know calculus &/ already have a solid grasp on mathematical thinking.
Yaakov A Sternberg I agree
@@yaakovasternberg6295 I disagree. These videos were what I used to learn these concepts before I learned further with Khan Academy videos. Starting with visual intuitions made the math easy to memorize as it made sense.
That visualization of the product rule is simply amazing. Once again math shows that complex things are made up of simple ideas...
Or more just that complexity is a trick of the mind. It all depends on how you view things.
Unless you actually know the proof of the product formula which is even easier than the geometric intepretation.
Stephen Sinnemann.
It is amazing to learn calculus with real interpretation and understanding, and not just repeating formulas...
I am very thankfull for 3blue 1 brown, because his explanaitiona are extramly clear and take you back to basic thinking!
Are you the anonymous professor at the beginning? 🙃
@@GothicKin it isn't
This whole playlist is gold, you deserves a statue.
Also, I was wondering which program do you use for animations, since they do math stuff that's probably not easy to achieve with a normal animation program. Googled it, found that you wrote a python library yourself to achieve this, the respect bar goes higher and higher.
yep! he made a python library which is open source, to make his youtube videos.
its open source as well
It's called Manim and is an open source animation software
Yes this is an impressive and very useful work. As a teacher I share it with my students. I find that I am jealous of them since I didn't have wonderful resources like this when I was learning Calculus.
I'm elated to find out that no one correct the grammar error. That genuinely makes me very happy.
I think I'm the last to realize that there are 3 blue characters and one brown character.
I am the last one for now.
holy fuck
Vahé Tildian I am the last one.
welp
i guess i'm last now
Matt S. i noticed thanks to your comment.
This series, Essence of Calculus, is like the Feynman lectures on Physics. It's a masterpiece.
It is simply the way the subject should be taught.
Congrats, I love this series
Please don't stop being. I'm in university at the moment and even though I thought I had a good grasp of mathematics because of good grades, nothing excites me more than seeing something explained properly, seeing the layers of abstraction erode away and leave behind that one little piece of truth. You are hands down the best educator on RUclips right now, Brady, CGP Grey, Crash Course, VSauce, they all wish they could teach like you can.
He's great, i love his vids too but dont bitch abt other youtubers
Heyy, Vsauce, Michael here
Patrick Jmt is pretty solid
Everyone wishes to be like Michael.
vsauce is solid.
Wow. It's incredible. After 5 semesters of studying applied math at university I am very familiar with all those rules. But never in my education have the concepts been explained that well and rigorously. I thought going all the way down to the fine details and reasons would be boring but it's actually so much more interesting than the "standard" way to learn calculus.
Nothing in these videos is rigorously. Its whole purpose is exactly the opposite, being intuitive at cost of the rigour.
I've fallen in love with the π character
Me too!!!
π
😂😂😂
same, it's really problematic
Me too😅
Me too
**TIMESTAMPS TABLE**
0:05 Initial quotation (chain rule like onion)
0:15 Last videos were about simple functions
0:30 What about derivatives of more complex functions?
0:50 There are 3 ways to combine functions: sum, multiplication and composition
1:00 Subtracting and dividing are special cases
1:50 Derivative of the sum
2:15 Example sin(x) * x^2
3:00 df is clearly the sum of dg and dh
4:00 Performing the division by dx we get to the formula
4:15 Derivative of products
4:20 Visualization of a product as an area
4:47 Each side depends on a function
5:30 Now let's analyse how a change of dx causes a change of area (df)
5:40 df is the sum of 2 rectangles (+ an infinitesimal of second order)
6:30 Remember that dx -> 0
6:45 Working out specific example
7:20 General
7:30 Mnemonic Left d-right Right d-left
7:55 May be strange if arbitrary but now you know the rectangles areas
8:20 D k*f(x) = k * D f(x)
8:45 Derivative of function composition
8:50 Example sin(x^2)
9:07 Another visualization: 3 number lines, x, x^2, sin(x^2)
9:45 Let's analyse a tiny nudge dx
12:00 Derivative of the outside with respect to the inside * Derivative of the inside
12:50 This is called the "chain rule"
13:05 What df / dh means
14:10 The simplification of dh represents a fundamental concept
14:40 Now you have the 3 basic tools
15:00 Please do practice calculations
15:40 Patreon supporters
15:48 Ads
+ErikCR Added ads timestamp, I do not think it is possible to be precise to the second with these kind of time-stamps (when do you link? When he says the word or when he starts the phrase or when he start the "paragraph" or when the image is displayed, or when the formula is displayed?), and even if it was I do not think it would be so important
Also there is a trade-off between practicality of time needed and precision, like this it takes me 30 minutes, second-precision possibly much more
+
Riccardo Riglietti small tiny change in time at second precision is neglible appreciate for putting it together
Riccardo Riglietti Thank you very much for putting the timestamps table! It helps a lot! :)
The hero we never deserved, but the hero we needed
Thank you for this entire series.
As a retired engineer I really enjoy reviewing the 'how' and 'why' of so many memorized principles used in my profession.
Keep up the great work!!!
The Taylor series comes to mind.
I got a A in calc 1 and 2 but your videos are really making me understand the difference between memorization and understanding.
At last, a refreshingly different way to look at Calculus. The more I get into this, the more I am in awe of Newton and Leibniz.
this us BY FAR THE BEST calculus series for beginners!
For beginner's?????😂😂😂😂😂😂😂
Videos can't be understood by beginner's.....
The best part is you got 197 likes😂😂😂😂😂😂😂
One of the best for a new look at calculus too.
Just because he talks like we should understand what he is saying, doesn’t mean it’s for beginners. Actually it’s very difficult.
Well, I am an eight grade student learning calculus from youtube.
No, in my opinion , these are the best for graphics and animation (I appreciate that thou) . Jim Fowler is also a nice channel ( mooculus )
I like to call the chain rule "The Russian Doll rule". I find that gives my students a better sense of how to "unpack" the functions when differentiating. You can takeout the inner doll while it's still inside the outer doll.
This is a great series, by the way. Looking forward to tomorrow's video!
I liked your way of thinking about it.
It also gave me an idea on how to explain recursive functions in programming to beginners in the area.
Yes great idea !!
@@rodolphov.santoro8829 The best example for recursion is folder in folder in folder. Ask students to write a function to parse path name (only introduce recursion after they know how to deal with strings), and they'll get the point. Fibonacci is the worst example, because you can do it with loop and easier. Don't use that, even if all textbooks do that. Your students will think recursion is just another useless pedantic nonsense.
@@y.z.6517 To complete beginners folder inside a folder is already a new concept.Though i agree to cs students it's a good idea, it's not to every public.
Some people like examples with real life objects, and it's specially usefull to someone who barely uses their computers.
@@rodolphov.santoro8829 "To complete beginners folder inside a folder is already a new concept." Technically true, but anyone who can learn coding already has lots of experiences with using computer(s). I agree that russian doll is a good example, but it is non trivial to translate that to codes. You need to know how to do animation. Then, you need to know relatively advanced maths like matrix and transformation. In practice, either a student gets it as a metaphor, or he is confused.
This is hands down the highest quality content on calculus I've found. I sent this to my old calc professor and I really think he'll enjoy it! Can't wait to see these every day when I get home from class. Keep up the amazing work!
Wish more math professors told their students why these things work and what they derive rather than reading straight from the book and writing endless examples on the board. Thank you for this amazing video, your the math professor I never had.
Just sitting down and watching each of these videos has taught me more about Calculus than all of my years of college calc.
I've lamented the fact that for me Calculus has been little more than a series of steps I go through to solve a problem rather than a process that I use to think through and understand what is what is actually going on.
This video series in a single viewing has helped me more than anything else to really understand Calculus. I really look forward to this series moving on to discuss Integrals.
Thank you so very much.
If you hadn't taken years of calculus, you would not have any idea what this video was saying!
@@lwh7301 I've not done years of calculus, and I'm able to understand what's going on here so I don't know what to tell you
@@velyro4424 Well, aren't you special?
@@lwh7301 Similarly to @Velyro, I understand the video without having taken any calculus.
@@moltenmath Well, aren't you special.
And here I thought Brilliant gives you a good insight...
I simply cannot express with words how useful this series is. Simply the best. I wish there were textbooks written like this.
inb4 animated textbooks
are you going to cover l'hôpital's rule ? i've never really been able to visualise it
Minh Dat Ha He is covering calculus, not so much limits (I know, limits are necessary to calculus, but I would not bet on this...)
+Minh Dat Ha Yup, in chapter 7 on limits.
Ah great thanks :)
This one was very intuitive btw. Keep it up!
Why call it Bernoulli's?
When the best quote is from an anonymous proffesor..
I'm about to finish up my bachelors in mathematics, and it's so crazy to see these things that I've known and used for years as just a formula plug in to be visualized and fully explained. Thank you so much for this video series, reminds me of why I fell in love with math in the first place :)
I'm so happy I found your channel. The feeling of being able to enjoy math again is somewhat overwhelming. Your method is spot on.
Was lucky enough to have a teacher in high school who explain the chain rule and other calculus topics the way they are here. Not as visually appealing, but with the same inquisitive approach to learning. It makes a whole lot of difference on having kids love the course and not dread it.
This is something that high school students need to watch before they enroll into their university, especially when you are in math-related course. There is nothing more scarier than not knowing the basic essence, or mechanics of what you are doing in uni. Here I am uni student majored in economics, learning calculus AGAIN in my holiday
I think watching this video will scare any HS student from enrolling in math classes. This is beyond what a typical HS student can comprehend.
I've been trying to learn calculus and specifically the chain rule for YEARS, and haven't had the motivation and time to do it. Today I finally understand it, in no small part thanks to these videos which made these concepts intuitive, elegant and exciting. Thank you so much!
the quote at the start , that putted a smile on my face
I can't tell you enough how much I love this series, actually understanding what is taking for granted in math classes feels amazing and it has helped me tackle much harder problems (e.g. in math competition). You are basically my main educator when it comes to math and it is really helpful. Love what you are doing, thank you so much!
Knowing how math works is so much better than learning formulae in class. Not just because it gives you more knowledge/perspective but also from an exam perspective where you can come up with the formula by yourself because you know why/how it exists. Thank you so much for making this amazing series!
I just recently discovered this series, and I think it's cool that you guys explain why something is true, rather than just saying "take it as true".
This series is awesome. I struggled with calculus in college, but your approach here has made things so much clearer!
I watch this series night by night. just constantly amazed by mathematics and your way of telling stories. it’s like watching a fine true art.
This channel has made me fall in love with Math and changed the way I used to look at things, very much appreciated and thank you for taking so much effort to create such a discriptive videos. I have a mind of a Mathematician, keep on the work of teaches others.
The product rule of derivative is so enlightening. Never knew, neither imagined! This is not just calculus. This is more like a realization through some philosophy! Pranamam GURU (Namaskara in a deepest, revered sense)
5:50 holy moly it all makes sense.
I've been learning for the last weeks for my phd defense and watched a lot of videos, but you are really on top of all learning sources: best explanation, best animation, easiest understanding..! Please carry on and thanks a lot!
These are truly beautiful videos. Thank you so much. I find your explanation style crystal clear - the animations are insightful, elegant, and efficient. It'd be difficult to overstate how much of a gift you are to people who are seeking strong intuition in math. Continue, continue, continue making these please.
sir i havent seen a best teacher like u of math in my whole life .....hats off sir ....
Wish these videos came out 2 years ago when I was first learning calc!!!
Very nice videos, very intuitive and fun to watch!
I want to congratulate you and your team for this amazing visual work you do. These videos are incredible. Keep it going!
I couldn't understand the derivative of the quotient of two functions.
I asked for help to someone who ended the university successfully. That person spat more formulas that neither of us could understand.
Now I realize how important your work is for someone looking to learn calculus, not only solve calculus problems.
Thank you from Perú
The channel MindYourDecisions has a video showing how the quadratic formula can be derived geometrically. After seeing that, I was really interested in how it could be applied to other areas of math. Needless to say, your channel is the absolute best at this. I work as a financial auditor, so I deal with math everyday, but no more complex than basic math usually, but this channel and a few others really make me want to get into deeper math concepts. Thanks for all the hard work you must put into these, they are extremely captivating!
That (Anonymous professor)'s name? *_Shrek._*
For those who don't understand the obscure reference, 'Ogres are like onions!'.
What are you doing in my swamp
Drone_Better "obscure reference" lol you assume someone hasn't seen Shrek? ... **Donkey and Shrek laughing**
This series is ending literally the day before my calculus AP test, I can't thank you enough for making this!
I rarely comment on these videos but you sir completely changed my view on calculus. This is just pure gold. Thank you so much.
Only this channel can provide us with the geometrical and intuitional beauty of mathematics. Simply amazing!
That chain rule unfolding made me in awe.
Fortunately I got to know about you through "DOS",and now I'm marvelled with your thoughts , explanation , visualisation , narration and everything else.......
These videos remain timeless, and they are so apreciated. Thank you. Not only to you explain the topics in an easily digestible way, but you also teach math as something to marvel at and something to love. You make math fun!
I know @3Blue1Brown is probably not going to read it, but I just want to thank you for everything you have created.
You are a totally different vision of maths, capable to make anybody (even if they say "they don't like maths) being surprised and impressed of how 'magic' they can be. Those demonstrations, as well as the graphs and the fantastic explanations, turn you into like "woah, I think I'm quitting life and I'm going to do maths instead".
On the other hand, you have created a nice, respectful community arguing about maths in such an educated way. Your explanations are so simple that practically anybody can understand them. Therefore, everybody can contribute and even somebody who is 15 can point out something a 50-year-old teacher had never thought of.
So yes, thank you very much, and can't wait for another video.
Grant, once again, well done. I personally learned the chain rule as looking at gears and how the change of one gear changed the next which changed the next, but I also really liked your explanation. keep it up!
Yeah like a planetary gearbox!
All of these videos have been so helpful but this one in particular helped me over a huge hurdle. I had to rewatch about 6 times over a few days but I think combined with the homework I'm getting a grasp on it.
Thank you so much.
I wish I was taught mathematics like this when I was doing my schooling.....hands down the best video series on calculus. Better learning website comes close but only for specific things and not for general calculus like this. Kudos to you for creating this series. I am now many years past schooling but my love for mathematics brought me to this and am enjoying it thoroughly!
Your videos are second to none in their expression of mathematical beauty. I look forward to all of your future creations.
This channel is a goldmine of knowledge and understanding!
You are doing an incredible job. I've never seen such a 'plastic' view of the mathematics, as you do- this helps me a lot in understanding. I am really impressed and proud, that I found your channel on YT!
this channel deserves more subscribers, more followers, more people should know this channel exist. This is my goal for 2020. to tell people about this channel. i am beyond grateful for this channel and this channel proves that math is fun when you understand it.
I've been looking forward to this all day!
Just realized from this video that perhaps the most powerful part of this series is that it connects fundamental concepts that you learn before calculus (algebra and geometry) directly to calculus. I had always thought and been taught that calculus is this crazy new way of doing math.
It's not, it's just the same way of doing math with a few nice conceptual tricks and nuances.
exactly. It's wierd
This "complex" idea of small changes and dt's and dx's and those rules. And yet, this can be solved algebraically
that's why when d(sin(h))/dh = cos(h)
h = x^2
dh/dx = 2x
dh = 2x•dx
dy/(2x•dx) = cos(x^2)
dy/dx = (2x)cos(x^2)
litterly subbing in and moving and we get the derivative
its been six years and i think of this series every single day. perhaps one of the best to explain the application and deeper understanding of calculus
I just ran across these videos. I haven't had time to watch them yet, but I can see what the
approach is which is exactly the kind of thing I like. Thanks!
...................... Leibniz Notation now suddenly makes a whole heck of a lot of sense thanks to this video. I think when I go back to university, from now on when I work out derivatives I'm going to include that little dx or d[whatever] at the end to help me remember what's actually going on.
And this also makes the derivative identity d/dx cos(x)sin(x) = cos²(x)+sin²(x).
Trig identities have always been my Achlles' heel ever since I took precalc in high school, which I owe to a terrible precalc teacher who had an "I'm retiring" no-care attitude and an even worse textbook that only covered bare basics and then threw ugly curveballs in the problem set. A video series about how those work would be AMAZING and SUPER helpful to someone like me.
derivative of sinx*cosx is not sin^2x+cos^2x=1. It would imply sinx*cosx=x+c which is absurd. Derivative of sinx*cosx is cos^2x-sin^2x
I was initially going to study maths at uni, but I switched to Eng lit because my heart wasn't in maths anymore.
Now that I'm finishing a very fulfilling Eng lit degree, I have no regrets, but I do sometimes wonder what maths would've been like.
These videos make me feel like I'm doing a little bit of uni maths.
What job can you get with a degree in English or maths?
HarryIsTheGamingGeek Besides teaching, there's various positions in e.g. industry and business...
... And ALSO increasing the human store of knowlege, enriching the human experience. Money allows you to live, but science and art allow you to enjoy it. It's not just about jobs.
This is crazy, in uni right now ,just stumbled upon this series...binging this stuff rn 2 weeks before my math exam and I am just amazed how this guy ties all of the stuff together with such beautiful elegant explanations
Every video in this series blows my mind. I wish there had been time to go over this intuition when I learned calculus.
I knew calculus, but I didn't *understand* calculus, until now.
Anyone know a physics channel as good as 3Blue1Brown is in math ? I want something that can make me picture things like Gauss' law in my head
viascience. He's made introduction courses on general relativity and quantum mechanics
Thanks i'm going to check it out
I liked the Yale lectures with professor Shankar.
Veritasium
Jean Panachay Gauss' theorem actually comes under mathematics. We just use it in physics.
By far the best math channel on youtube. 10/10 wish I could like and subscribe a hundred times over
This is the highest quality lesson I could hope for. Thank you for the amount of effort you put into these explanations, it truly is unparalleled.
Your videos really help me with understanding the uses and applications of calculus in a real-world setting. Thx for doing what you do.
You always make these videos at perfect times for me
Grant, I am 12 and I have always been fascinated by math. Other than my dad, I have not found someone who teaches math well. I love you videos! Thank you for all you have done!
I love that you've turned multiplication into a great tool! Back when I was taking these calculus classes and struggling to "get" the math, I felt like my inability to visualize anything other than multiplication meant I didn't understand any of it. Whereas in this video, you demonstrate the flip side: leverage whatever you've got to get yourself further.
Thank you for this series. It really makes all of this so much easier to understand.
this series literally are giving me goosebumps. i feel enlightened, it's like i've been given the forbidden fruit.
jesus
your videos are necessary for the world, thank very much!
I love your channel, you gave a through explanation even the smallest details.
Thank you so much for this series! This has helped me understand calculus on an intuitive level, not just memorizing formulas and patterns. This is awesome, you're awesome, and keep it up!!
Hmm... and I'm here thinking, "Wow, this makes absolutely no sense whatsoever. I'm so glad I can just memorize formulas and patterns. Way easier that way!"
I've never had so many "oooooh, I get it now" moments before
Something that'd be great is to have some sort of little timer in a corner when we're supposed to stop and ponder. That way one knows when to pause before you move on.
Amazing series, thanks a ton for spreading awesome knowledge in an awesome way!
Exceptionally exceptional clarity in concepts. If wish I had found you in 2000 . All higher engineering , science topics use to go over the head due to poor grasp of math topics. U r doing exceptional service to academics as well as to future advancement of science and technology
i discovered ur channel yesterday and i love u
i always thought I'm quite good at explaining math. very good perhaps. but thats still heaps and miles away from 3blue1brown-good..
your visualization of these rules is incredible
Reading the comments here, I'm happy that at least I found this series before any formal calculus education. We will start basic calculus after some 6 months.
0:00 intro
0:49 three basic ways to combine functions
1:50 Sum rule
4:10 Product rule
8:41 Chain rule (function composition)
14:36 outro
"first D second plus second D first"
that's how I learned it
This video series is not only brilliant, but more importantly it's also inspiring. Sincere thanks.
Dude, this series is BRILLIANT! Thank you forever!!!
@3Blue1Brown Thank you for your passion! How do you create these nifty visuals?
I reaaaaaally wish I had this when trying to learn calculus
Well, it's been a while since you posted this video. Just wanted to leave my thanks for carrying my through calculus lessons.
You have my eternal gratitude, friend.
These visuals and your explanations are unparalleled.
will you be my math teacher? srsly, i learned more from your video then a whole semester with calculus
I can see your videos all day
He explains the things we have memorized until now in a very clear manner. Thanks a lot.
THE BEST EXPLANATION OF THE CHAIN RULE!
Thank you Grant for this amazing wonderful channel - spreading knowledge to the masses in such a revolutionary way of teaching. I hope you continue getting the support you deserve for your great work. With regards to this video I have one question. You mentioned that `d(x-squared)` would evaluate to 2xdx. That makes sense as you explained this in the last video of this series. However, I didn't understand why `d(sin(x))` would evaluate to `cos(x)dx`, as in why multiply by `dx`, as part of its derivative? In your last video when you explained how the derivative `d(sin(θ))` becomes `cos(θ)`, but there was no mention that it's also multiplied by dx! A clarification over why the derivative of `d(sin(θ))` is `cos(θ) mutiplied by dx` would be really appreciated. Same question goes for why d(sin(h)) is equivalent to cos(h)dh (i.e. it is multiplied by dh)? Many thanks in advance.
This part confused me also.
As Grant points out, the derivative of sin(x) is cos(x)
Therefore d/dx sin(x) = cos(x)
Thus d(sin(x)) = (cos(x))dx
What I don't understand is how is d(sin(x)) is "equal to the change in the sin graph".
Can someone please explain this statement?
@@rhyzza459 Bro it is in the term itself. The d in d(sin(x)) represents the change in the function sin(x) and consequently in it's graph.
You're brilliant! Thanks for sharing your passion for math with us. Every time I watch your videos I feel more willing to study math.
you just can't get as quality an explanation like this on a chalk/marker board. fantastic review.
I liked a video for the first time in my life! You absolutely deserve it!