Derivative formulas through geometry | Chapter 3, Essence of calculus 

How is this kind of content free?! Respect man. Seriously.

I keep getting these "OOOOOOH, I See, so that's why!!!" moments while watching this video. this is great.

I don’t know words to express how grateful I am for 3b1b and Khan academy

We live in an age where a highly motivated individual (with internet access and time) could learn just about anything with no formal education. I hope this playlist stays available for a long time because it clarified so many things I wondered about and couldn’t articulate.

Dear first time calculus learners, Do NOT expect to understand calculus after one pass through this video series. You must "pause and ponder" a lot, draw pictures, and see what new formulas you can discover through geometry. Read your textbook, listen to lecture, and do your homework problems, and make sure to give the sections in this video a pass or two or three (or twenty). Calculus is amazing and wonderfully intuitive, but was not invented in an afternoon, and there's a reason that the course is two semesters. If you fully understand these videos and can do computations and solve word problems, it is safe to say you have mastery over the material. Good luck and enjoy learning this beautiful subject!

**TIME-STAMPS TABLE** 0:06 Initial quotation 0:15 How to compute derivatives? 0:30 Why is such computation important? 0:45 It is abundant in real world 1:15 Important to always remember the fundamental definition of derivatives 1:45 D x^2 example 2:00 Graph analysis 2:40 Graphical intuition for Dx^2 3:30 dx^2 is negligibly tiny 4:20 Algebra passage to obtain derivative formula for x^2 4:45 D x^3 example 5:10 Delta volume of a cube 5:40 Negligibable parts 6:50 Pattern for Dx^a = a*x^(a-1) 7:30 Usually just symbols, but why? 8:10 We can ignore much of the terms in the computation 8:40 General case of x^2 and x^3 9:50 The importance of remembering the why 10:10 Example D 1/x 10:20 (You could just use the power rule) 10:45 Geometrical interpretation 12:00 Exercise for the viewer 12:30 Now figure out D sqrt(x) 12:40 Trigonometric functions 12:50 Geometrical view of trig functions 13:35 Starting by looking at the graph 14:10 D sinx should be cosine based on valleys and peaks, but why exactly? 15:30 Demonstration based on similar triangles 16:50 Now what is D cosx?

Oh my god. When I first saw this video at the start of college in my engineering course, I didn't have any clue how to solve the 1/x and the sqrt of x derivatives via geometric analogies. Now that I quit my engineering course and am pursuing a computer science degree, I finally solved it after 5 years. I finally figured out the tricks needed to solve both equations once I got comfortable with the concepts behind calculus. It was a roundabout journey for me. I know no one will read this, but I just wanted to share. It's a happy moment for me! Thank you 3b1b for this series.

the derivative graph of sinθ is literally mind blowing. two years of calc and it finally makes sense. thank u for giving me hope for my ap exam in a couple days, this content is incredible.

Our math teacher shows your videos in class!

Thank you for existing

ah.. yes.. mhm of course! .. *goes back to first video*

I am a head of mathematics at a school in the UK and try my absolute best to teach my students and embed this sort of level of understanding. The one tool I just wish I had is animation! These animations are so clear. I use Geogebra to the best of my abilities but just can't quite offer the same visualisation as you do with these. What do you animate using? If I could just do animations a tenth as good I'd be happy. This level of visualisation adds that extra dimension for students to grasp a concept. I am very appreciative of your videos - once I have reached the limit to which I can explain something I show these videos in class to add that extra visual aid. So pleased to have your videos to complement my lessons.

Needless to say, the absolute _best_ math channel on RU-vid, not even close

So many educational videos on RU-vid are "edutainment" designed to give the illusion of learning something new, without actually teaching anything. This channel bucks that trend and I am SO grateful for it. Please never stop (or at least keep going for a really long time). Personally, I hope you eventually get into the math behind some concrete practical applications like machine learning algorithms, but I'm loving these pure math series too.

15:23 when he switched voices it kinda scared me loll

Watching this series has really made me wish 16yr old me was as motivated and appreciative then as I am now at 33 of how interconnected the various maths are. I literally had a flash back to highschool and had a legitimate "ahaa" moment. This is truly excellent content!

Next up will be "Visualizing the chain rule and product rule": ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-YG15m2VwSjA.html You’ll notice throughout this series that I encourage a more literal interpretation of terms like “dx” and “df” (aka differentials) than many other sources. I call this out and explain further in many of the videos, especially chapter 7 on limits, but given that students are often told not to take these terms too seriously, to be wary of treating them as literal variables, it’s probably worth adding another comment on the matter. The path between treating these terms as literal nudges and a fully rigorous treatment of calculus is actually quite short, considering the loose language that seems to be involved. You just need to understand two things that are implicit in the notation “df” and “dx”. First, the size of the nudge df is dependent on the size of dx. It is not its own free variable, and what it means depends on your current context. Second, for any equation written in terms of df and dx, when you replace dx with an actual number (e.g. 0.01), and replace df by whatever nudge to the output is caused by that choice of dx, the equation will probably be slightly wrong, with some error between the left-hand side and right-hand side. But what it means to be using these differential terms is that that error will approach 0 as your choice for dx approach 0. This is why terms which are initially proportional to (dx)^2, and hence retain a differential term even after dividing by dx, can be safely ignored. Even in the most rigorous proofs of derivative rules and properties, these tiny nudges show up, though often under the names "delta-x" or "h". The ideas presented here are essentially the hearts of those proofs but phrased without the surrounding formal language. I put together this series not just with calculus students in mind, but also with the hopes of pointing back to chapters here when I cover real analysis, the formal backbone of calculus, so I am motivated as much by an ultimate desire for people to understand the rigor as anyone else. (Also, as a hint to those asking about how you know that the triangles at the end are similar, use the fact that the tangent line of a circle is perpendicular to its radius.)

These videos are art... Really, they are simply works of art...

For those who are not understanding this, just keep rewatching this video and do not give up. Even Im going for a 4th rewatch and now it seems that im starting to appreciate its beauty!!

For the f(x) = √x case, the reason why the new area is represented by dx and not df (as in the x^2 and x^3 examples) is because we square both terms in f(x) = √x to get (f(x))^2 = x. The blue area is therefore f(x) * f(x), which is simply √x * √x = x. The new area, dx, is created by a 'nudge' df(x) in both directions, which is just d√x. From there dx = 2 * √x * d√x + (d√x)^2. Ignoring the (d√x)^2 terms since they go to 0, you get d√x/dx = 1/(2√x).

Hello! I am from Brazil and would like to thank you for your work, I am a student of Industrial Chemistry and in my country we have a bad basic education, at the time there were no platforms like this, preventing access to content like yours. Thank you so much for dedicating your time to the cause of education. It is very important to many people like me.

I just started learning calculus. My math teacher taught me some formulas but when I asked him "but why?" he didn't really have an answer. Until I came across this channel I had many questions. I'm really loooking forward to next chapters. Keep it up.

I wish I could have watched this video 30 years ago when I was studying calculus.

Now i am no more going to give any mathematical exam, but i loved watching you videos , i wish you would've present when i was in school.

I took calculus almost 6 years ago now. I'm now a grad student in robotics and diffeq is life. I love seeing how some of these things come about that either: were never explained to me or had been forgotten due to the years of plugging away.

If anyone's wondering what the justification is for the claim he makes at 15:39: The base of the small triangle is perpendicular to the right side of the large triangle. The hypotenuse of the small triangle has a slope very close to the tangent of the circle at angle theta, and therefore is roughly perpendicular to the radius shown (the hypotenuse of the large triangle). Thus, the two angles of those two triangles that are touching are about the same. We also know they are both right triangles, so that's two angles that match. There's only one possible value left for the remaining angles (sum of interior angles of triangle = 180 degrees), so all the angles match, and therefore the two triangles are similar (well, mostly, but they get more similar for smaller values of d-theta).

For people wondering how d (cos θ) = - sin θ Note: While moving around the circle, sin θ is increasing but cos θ decreases from 1 to 0 and then continues its simple harmonic motion. Just use that line of reasoning and you can see at 16:56 that derivative of cos θ is - sin θ.

This is so good. My second time watching and this time taking notes and drawing some of the diagrams. I am so grateful for you sharing your experience.

Here is my solution to 12:21 Area gained + Area lost = 0 Area gained = (1/x - d(1/x))*dx Area lost = x*d(1/x) Adding the areas x*d(1/x) + (1/x - d(1/x))*dx = 0 "Distribute" the dx x*d(1/x) + (1/x)*dx - d(1/x)*dx = 0 Rearrange to factor out d(1/x) in next step x*d(1/x) - d(1/x)*dx + (1/x)*dx = 0 Factor out d(1/x) d(1/x)*(x - dx) + (1/x)dx = 0 Subtract (1/x)dx from both sides d(1/x)*(x - dx) = -(1/x)*dx Divide both sides by (x - dx) AND dx d(1/x)/dx = -(1/x)/(x - dx) Distribute the terms in the denominator on the right hand side d(1/x)/dx = -(1/(x^2 - x*dx) The second term in the denominator on the right hand side will go to zero as dx goes to zero. The solution is: d(1/x)/dx = -(1/(x^2))

I'm a math major currently finishing up my second semester of Advanced (i.e. proof-based) Calculus. I just learned more about why D sin(x) is cos(x) than in all my years of math up to now.

He’s explained so many math concepts better than any teacher or professor that I had. I took calc 1 and 2 but never was able to fully grasp what derivatives are, how they work. This video did explain it so well.

Taking a step back to remember why the power rule works is literally why I'm watching this series, so thank you!

That explanation of the derivative of sin at the end is mind blowing. Thank you for making these video, they're so well produced and written.

Your reasoning of the derivative of sin(x) was beautiful. One of the nicest connections I've seen.

I'm on a rewatch of this series, and wow! This episode is still mind-blowing

15:50 the reason that "little angle" is equal to θ is because the hypotenuse of the small triangle is considered a straight line, and therefore it can be considered the TANGENT of the circle. Since it is the tangent, it is perpendicular to the radius of the circle, and the rest is now obvious.

An "Essence of group theory" series after this one would be awesome

This is a very good video explaining the reasons behind the basic rules of derivatives that school rarely or never teaches. Great job!

37 years since calculus in college...lights go on with this simplified and better way of teaching.

Your channel is absolutely beautiful work. Of course there is much more to each of the subjects they approach, but when used in conjunction with more standard Mathematics pedagogy, your videos enable so much deeper understanding. Thanks so much for what you do.

Oh geez I did the viewer challenge! For once I actually completed a viewer challenge! I know people are gonna think I'm dumb for finding that "breakthrough" profound, but I did a viewer challenge!

This is all so simple yet so profound. I love rediscovering calculus through non-hostile eyes. the whole animation involving 1/x was so elegant i loved it

As we were thaught to be in the Future now addicted to our historical Calculus times, this is because of you, Many thanks for your efforts, this is a great math channel and I'm recommending everyone!

I struggled through calculus in college back in the 90s. These videos are simply fantastic and provide so much better understanding of the why vs just memorizing things.

Have you thought about doing more videos over complex analysis?

I hope you get a prize or something for what you're doing. It's incredible

Best tutor everrrrrrrrr I am a Biology Olympiad participant and I needed a good comprehension of derivatives and integral for statistics, population ecology, probability and physiology topics which I accomplished with this channel's videos. Thanks a lot. edit: I'm Iranian and I'm aware of the lack of fluency of English and accessibility to RU-vid among Iranian students. I would be grateful if you give me the right and cooperate with me, so I can translate your tutorials and share them with my friends.

Well done! I have finished my first half semester in Calculus 1 at a private University and haven't learned as much in 2 months as I can in 2 hours of listening to these fastidious explanations. Well made!!!

I have never appreciated the beauty of derivatives up until this video...thank you so much!

I've been applying the Power Rule so many times ever since I learned about derivatives in calc, but never truly understood why the formula is the way it is. After seeing the geometric visualizations for x^2 and x^3, it makes a lot of sense now. Thank you for making these videos, seeing all these different interpretations of formulas I didn't give a second thought about is really enlightening. I look forward to the next 7 days of videos.

This is going to be a great introduction for my competition calculus. This will make some of the abstract concepts of this subject appear much easier on those tests. For that I have to thank you

Thank you for this video. I love math and this is everything about Calculus my math teacher doesn't have time to teach us because of the demands of the schedule, but it's also what makes it profound. Keep up the good work. Abrazo!

OH!!! I finally understand why the triangles at 15:39 are similar! (Maybe I'm dumb but I saw lots of people in the comments with the same question.) I don't know if I'm able to explain it on a simple written comment but I'll try. First of all, let's only look at the big triangle. Its internal angles are θ, 90° and the other one which I'll call φ. I hope you know by now that θ+90°+φ=180°. That's a property of triangles. I'll also call the big triangle BT and the small triangle ST. The radius of the circle is always perpendicular to the tangent at that point where the radius touch. Therefore, from the radius to the tangent, there's always 90°. That's pretty obvious, I know. But it wasn't obvious to me why that was important. The angle from the BT's hypotenuse (or circle radius) to the bottom part of the ST must be θ. I hope you can see why because it's hard to explain... One of many ways to see it is to notice that the angle between the bottom part of the ST and the right side of the BT is 90°. 90°-φ=θ. Because the ST's hypotenuse is colinear with the tangent, we know that the angle between the ST's hypotenuse and the BT's hypotenuse (also the circle's radius) is 90°, as told in the first step. Now we can find out one of the unknown angles on the ST. θ+(Unknown Angle)=90°. So (Unknown Angle)=φ. If the ST has φ and 90°, it must also have θ. (θ+90°+φ=180°) If both triangles have the same angles, they're actually proportional. This was probably extremely confusing. lol Also, my english is kind of rusty, so sorry about that. But I think one can understand the problem if this is read carefully.

Your videos are concise, entertaining, and poetic. I'd love to see the Essence of Probability series!!

A better name for this channel - pause, ponder and rewatch!

I cannot stress enough how helpfull this has been. Going through Highschool and Uni where only surface level explanations are given can dissolution you and make you forget why you ever liked math and science in the first place. These videos are helping so much to re-ignite my curiosity and remind me why math and science exited me so much in the first place.

15:39 why the triangles are similar (commenting so i can look back at this, except im not a big brain math genius like everyone else here) - big triangle angles: θ + 90°+ (other angle)= 180°, so θ + (other angle) = 90° - radius/big t's hypotenuse is perpendicular to tangent line of circle (hypotenuse of small triangle) - knowing that alternate interior angles are congruent, angle btwn radius and bottom part of small t is θ - because angle btwn tangent line and radius is 90° (hypotenuse of small and big triangle), 90°- θ = (other angle) - this means that the far right angle of small t is "(other angle)" - because small t has a right angle and has (other angle), and θ + (other angle) = 90°, the last angle is θ. - because the angles of both triangles are the same, they're similar

I’d love to say a huge thank you to you. All the videos you have made are absolutely fascinating and beautiful. I remember being so deeply moved by maths when I first saw your topology videos. They have motivated me a lot to pursue mathematics in my further studies and I am so glad to have you to be my best maths teacher. Don’t stop making videos and thank you very very much!!!

This is really mind blowing. Never got these insights from my teacher. I am really grateful for looking at this video and the channel. Keep it up and keep inspiring

I was really struggling over conceptualizing the derivatives of trigonometric functions until I watched this video. No more just remembering what seemed like arbitrary rules! Thank you so much!

you are math god. It took me years of study and even more research to understand the essence of math. Wish u existed 10 years ago :(. I had only one good math teacher in collage, but u outshine everyone. Your explanations are simply beautyful, intuitive and simple. When i was studying i had the same approach to math problems. PLEASE PLEASE continue your work. I would like to see you explain FUNDAMENTAL FORMS, FOURIER SERIES AND SPHERICAL HARMONICS. I had very hard time to understand those. I consulted countless professors and used Bronstein math manual, wolfram wiki, everything. Still those are still abstract subjects to me. Pls help

I literally have never seen(heard actually) a better teacher than you. You are actually helping us students alot by making these videos. I hope something really good happens to you someday.

Fantastic. The best illustration ever. ❤

I have just done math without understanding the basic concepts and visualization like this. Many students including me didn't find the right path to go through with amazing teaching before. I really appreciate this channel and you are my most favorite teacher., Of course pioneer too

I am a Vietnamese student, I can remember lots of derivatives but never did I understand their meanings. But only until I find out this channel, it's enlightening!

This is amazing. These videos are starting to be a part of the morning that I look forward to, 3Blue1Brown, I cant thank you enough!

I am fairly sure he is the greatest mathematician of our time. His ability to find the deeper truth behind common maths is simply brilliant.

That was so amazing, my high school teacher couldn't just gave these to us as formulas to memorize with no real logic behind it. I knew from watching ur other videos on derivatives that there must have been a reason why the derivatives have these exact values, but this video just explained even better than I could have imagined. Thank you so much ❤

Your videos are so beautiful. They really express the pure beauty and elegance of mathematics and also physical phenomena. Unfortunately, not everyone on the earth can or wants to experience this beauty. I am privileged. Thanks !

My favourite channel on youtube. Your efforts are so appreciated :) I would love to see a series of videos on probability.

The way you explained this calculus stuff is awsome. Thank you for putting so much effort to educate the masses..

This playlist is amazing! Thank you for creating this, so many fabulous insights.

Where were you when I was working my way through engineering school by rote; just going through the motions without understanding the underlying concepts. Great job! and a real service!

I wish you were my math teacher! I also wonder what an hypothetical series named "Essence of Trigonometry" would be ;-)!

It's my first time watching this channel. Wow, your explanation with visuals is so great! Crystal clear! Thanks for all.

Wow 😮 that’s exactly what I was looking for quite some time. Your series 1 was equally brilliant. Kudos for that.

Your videos have taught me to imagine a lot. I'm an aspiring data scientist and many of my friends follow your content. A request will be making such short series of probability and statistics series. Would be really helpful.

This video is incredible. I'm extremely thankful for your awesome content, 3Blue1Brown!

I just want to say a huge thank you for these videos. I have been studying derivatives in school and completely not understanding *why* the formulas my teachers tell me to memorise work, but since watching this series I have finally been able to see that there is a perfectly simple reason why e.g. the derivative of x^2 is 2x. I wish all maths in school could be presented with the proofs of why it works, not just given out with the instructions to memorise it.

I watched this video 2 years ago when I took calc 1, and I think it’s what really made me fall in love with this channel.

There is magic in your videos... concepts become crystal clear

You sir are a gentleman and a scholar. Your videos are absolutely amazing!

You have shown calculus in very different means realistic way, no one like us can ever imagine. Thank you very much sir.

As someone who just learnt Calculus as a pre-requisite for my Calculus Based Physics course, this really helps me understand the blackmagic of 2 stepping down in x^2

3:58 Maybe a better explanation than "this is so tiny, you can ignore it (nevermind the other term also gets truly tiny and will not be ignored)" would be to actually divide by dx once to get df/dx=2x+dx and let dx approach zero, so that df/dx approaches 2x. EDIT: You did actually explain it this way at 6:11 :)

My brain has to work so hard to wrap around this stuff, but when it finally does its so so satisfying.

Before I bumped into your channel, I had almost only algebraic intuition than the visual side. In order to make the algebraic process get etched into my intuition, I imagine that I was to explain those math concepts to some family members who were conventionally deemed as ‘have no mathy brains’, such as my brother, whose highest diploma is from primary school. And the reasoning process needs to be as plain as possible so that it fits Einstein's instruction to us: ‘If you can't explain it simply, you don't understand it well enough.’ This visual math induction of yours is somewhat like a superpower to me. And thinking about it like a superpower makes me wanna learn it. So I made watching your videos part of my morning routine. The surprising result is that now I can confidently say these two things: Math is fun. Getting to know math is NOT that intimidating. Thank you. Grant.

This is mathematical pure gold! Thank you for showing how abstract math is really related to geometric logic and intuition!

Watching these videos I can see how Math education in schools suck really really bad! Why the hell on earth we didn't have such methods and teachers to teach us Math. I could still use a teacher who knows math by heart and can teach even this is my junior year in University. I hope people who memorises Math and think they know Math, stop being teachers and leave some space for teachers like this.

These videos should be shown in schools not the pile of formulas.

Really appreciate how easy this channel makes to understand maths!

Man the nostalgia. I watched this a couple years ago and I still think it's the best introduction to calculus ever

I'm about 50 yo, all my life I was afraid of math. Calculus was a nightmare for me. With this channel, I feel like I've defeated my ancient fears. Thank you

Holy balls this series. I had one math teacher in school that taught us new formulas by going over how they are actually developed and I could understand everything perfectly, but after he left to teach at a university, I never understood what I was doing. I was just remembering how to use formulas. You have no idea how useful all of this is to me... I've already taken calculus and got ~65% in it, and it's part of my University course so I get to take it again and I've watched 2 of these and I already feel like it's going to be a piece of cake. If you are still around when I get a real job out of this course, I will repay my debt to you :))

I'm from China, and my words just fails to express my gratitude to 3B1B! I'm a student from data science, and I really needed it to refresh my memory ! Thanks a lot!

It's unbelievable... I have been looking for such information in a good presentation for two months

nth

For the case f(x) = 1/x: The blue area + red area (area lost) = 1 The blue area + green area (area gained) = 1 This implies the red area (area lost) = green area (area gained) Red area = -d(1/x) * x Green area = [(1/x) - (-d(1/x))] * dx = [(1/x) + d(1/x)] * dx Since red area = green area, we have: -d(1/x) * x = [(1/x) + d(1/x)] * dx Dividing both sides by x * dx, we get: -d(1/x)/dx = [(1/x) + d(1/x)] / x Ignoring d(1/x) on the right side since it approaches 0, we have: -d(1/x)/dx = (1/x) / x -d(1/x)/dx = 1/(x^2) Dividing both sides by -1, we get: d(1/x)/dx = -1/(x^2) Therefore, the derivative of 1/x is -1/(x^2). Power rule d/dx(x^n) = n*x^(n-1) works even when n = -1.

I loved that geometric explanation and how nicely it connected with the power rule for me. Finally I get it! Thank you

I was introduced to calculus 3 years ago and I have hated it since. I couldn't have believed it was this beautiful. I love you man!