The essence of calculus

What might it feel like to invent calculus?
An equally valuable form of support is to share the videos.

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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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:

Stream the music on Spotify:

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).

Various social media stuffs:

[avaneeshtisgaonkar]

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

[tomasforastieri]

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.

[avishkarn]

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

[SingingTurtlePress]

Thanks! I'm a math tutor, and I must say that 3Blue1Brown is a remarkably interesting, insightful series. Thank you, Grant.

[jawadtahmeed]

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. :)

[gotohetch]

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

[juancharovsky]

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!

[csstuff]

Coming to this video after my final class in my first calculus course is mind blowing. Spotting the connections earlier on would’ve saved me so much sleep. This is an amazing video thank you!

[michaellubrecht]

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!

[johnbarbuto]

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!

[drsteviejasengnsangma]

5 years have passed but this video still brings inspiration to many to love calculus even now❤️

[VibinAvic]

I’m currently taking calculus in high school and I can confidently say, they do not teach you how to understand the concepts like this video does. In ten minutes, I was able to fully understand the countless hours of plugging in numbers my teachers had me do. I am deeply grateful for this video.

[SweetComputing]

"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.

[bharathbhat]

"Math has the tendency to reward you when you respect its symmetry" - Grant Sanderson
My new favorite Grant quote!

[SachinGurjar-qqhg]

The quote "The art of doing mathematics is finding that special case that contains all the germs of generality." really touched the deep of my heart. Excellent deep meaning in that. Thanks Grant for the wonderful work that you put in

[drdrago]

as an engineering student working my way through an accelerated calc sequence having taken 0 calc in HS, this video is filling in the gaps for me with these visual aids. your answering all the side questions I wondered and small mistakes in my ideas as you go along. (and of course the prof is too busy to answer while they speed through problem explanations.)

amazing supplemental videos here! still holding up 5 years later. it's the high level concepts & gap-fillers tips/hints I knew I needed but couldn't find.

[Kaerulans]

This is probably my favourite moment of this week

[rileyj.s.]

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.

[rouey]

Wow! Wow! Wow! I'm an electronics engineer turned Software Development Manager over the course of 20+ years and I decided to go for a Masters in AI, mostly for intellectual interest in the area. So, I started revising my Maths when I stumbled upon this and was left literally speechless with the clarity, insights and clear explanations of the fundamentals of one of the toughest areas of mathematics. How so much fun and productive my uni years would have been had I had access to this type of free (!!!!) content. What difference it would have made in the training of engineers, many of which just developed strategies to squeeze through their exams, never having properly grasped the subject and its relation to their areas! Congratulations and thank you so much!

[hammyboye]

in my math class we have done nothing but algebra and geometry and when it came time for calculus, our teacher didnt spare any time for introduction. when i asked why the derivative of some function had that result, the teacher just responded: "there we go with those phylasophical questions". im not sure if the problem is in the teacher of the school system but either way im glad there are other sorces where i can learn and not memorise a chart