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The essence of calculus

3Blue1Brown

11,750,645 views

What might it feel like to invent calculus?
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In this first video of the series, we see how unraveling the nuances of a simple geometry question can lead to integrals, derivatives, and the fundamental theorem of calculus.

Thanks to these viewers for their contributions to translations
Dutch: @LFWarsen, Lauri Warsen
Hebrew: Imri, Omer Tuchfeld
Hindi: Vinayak
Hungarian: homok43
Italian: @Deye27, @hi-anji, alberto alessi
Korean: Hana Seo
Portuguese: redkk123
Spanish: @agustin-j
Vietnamese: @ngvutuan2811, ngvutuan

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These animations are largely made using manim, a scrappy open source Python library: https://github.com/3b1b/manim

If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/a...

Stream the music on Spotify:
https://open.spotify.com/album/1dVyjw...

If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended

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20 Comments

@Mepharias 7 months ago

I am currently a junior undergrad pursuing a degree in physics. I work part time as a peer tutor at my University's Tutoring Services center. I am not joking when I say that I recommend this series to nearly everyone who comes in for calculus and demonstrates or declares a lack of conceptual understanding.
This video specifically is the reason I went from hating math all through compulsory education and going nowhere in life for 3 years after high school to deciding to return to university.
I can say with absolutely no exaggeration that this video genuinely altered the course of my life. I get a bit emotional even thinking about it. So, from the bottom of my heart, thank you for creating this. You've contributed more to my success in life than any teacher I've ever had. As far as I am concerned, this video and series are the final word in conveying not just the ideas underlining calculus, but the very concept of mathematics as a whole.

@gotohetch 7 years ago

My calculus professor usually just says “lol here are some formulas and rules to remember”. Higher education at its finest.

@ChannelOneA 3 years ago

“Math has tendency to reward you when you respect it’s symmetry” - has easily became my favorite math quote!

@elamrbilal 1 year ago

Let's show Grant Sanderson how many of you are waiting for "The essence of Probability"

@avaneeshtisgaonkar8131 6 years ago

Can we all appreciate how much effort Grant puts into the animations in his videos so we can understand?

@angusk4932 5 years ago

this just proves that maths isn't boring, unenthusiastic teachers are boring. Literally 25 minutes ago I was in a lecture being taught this and almost falling asleep, watching this, it's literally like a completely different subject.

@SweetComputing 5 years ago

"Math has a tendency to reward you when you respect its symmetries."
"Mathematicians don't just care about finding the answer but developing general problem solving tools and techniques."
"Transitioning from something approximate to something precise is pretty subtle and it cuts deep to what calculus is all about."

As much as I love your clear cut explanations of the topic - these hidden gems of wisdom is what attracts me to the videos you create and helps me build an intuition around the subject which my university professors failed to. This is the sort of insight which is developed when you've spent enough time with the subject and examined it thoroughly from all perspective. I am very grateful of the effort you've put into making these series' and I will be coming back to it from time to time to jog my memory.

@0513.R4nDy 1 year ago

If you did not understand this video at first, watch it 2 to 3 times and you will definitely understand. Do so with patience and with an open mind.

@jwd.t 8 years ago

As a school going kid, I always wanted to visualize math like this animation. But my teacher could never provide with any material like this. Now, several years after finishing my bachelors I can finally feel these ideas for real. Thanks a lot for the hard work and this fantastic motion animation. I wish, in future you will cover each and every aspect of math, science and technology. The world needs these kind of content more and more for the people to better understand the underlying rules of our world. Cheerio from Germany. :)

@tomasforastieri9194 5 years ago (edited)

I'm a high school student from Brazil, and I can say that you've inspired a generation of people to pursue math and physics for their own sake. I started to learn calculus through these videos 4 years ago and now I'm studying for the AP Physics C exams (wish me luck lol). I know you'll probably never see this, but if you do, just know that we really appreciate your work.

@michaellubrecht7224 5 years ago

This video blew my mind. I took calculus in college 45 years ago, and it nearly killed me. Dropped the class a couple of times before finally squeaking by with a sufficient grade to satisfy my geology degree. I haven't had much need to apply calculus since, but my lack of understanding has always bothered me. Our professors just blew through it so quickly, I never had a clear understanding about why some factors were important and others could be diminished and discarded. Grant's simple explanation of the area under the curve equating to the piRsquared formula actually caused me a physical reaction! Why hadn't that been shown in any of my classes? Now, at age 67, math comes a little harder for me, but I still plan to work through this entire series!

@spiiegel23 11 months ago

2:25 -"AND MATH HAS TENDENCY TO REWARD YOU WHEN YOU RESPECT ITS SYMMETRIES"

@rileyj.s.5899 6 years ago

The reason I like math is because I’m obsessed with using logic to find answers rather than memorizing. That’s why I loved these type of videos.

@johnbarbuto9732 4 years ago

Brilliant: not just "what's the formula" but the deeper understanding of "how they figured out the formula"! This is the way to learn not only math but how to do analytical thinking!

@DrTryloByte 9 years ago

How does this guy make a 20 minute long math video that feels like it went by all too fast?

@juancharovsky252 3 years ago

I'm almost 30. Never in my whole life I had calculus explained to me in such an intuitive (and therefore effective) way. I'm watching the whole series and taking notes. Just priceless. Thank you!

@EagleSlightlyBetter 8 years ago

This channel is like listening to Mozart.

@Kaerulans 9 years ago

This is probably my favourite moment of this week

@ExistentialDodo 6 years ago

Me: "What's a derivative?"
My calculus teacher: "Derivative is Derivative!"

@Kashvi_Aayansh 5 months ago (edited)

I clicked on this video just hoping to get a better grip on calculus for my physics … but it turned into something I didn’t expect at all.
For the first time, calculus didn’t feel like a bunch of formulas to memorize , it actually made sense. The way many ideas were shown visually in form of animations helped me finally see what’s really going on .
It’s kind of amazing that even after 8 years, this video still teaches better than so many new ones.And honestly, thank you Grant. The effort and thought you’ve put these videos really shows. You didn’t just teach a topic—you helped me genuinely understand something I used to find confusing. That means a lot.