What's so special about Euler's number e? | Chapter 5, Essence of calculus

What is e? And why are exponentials proportional to their own derivatives?
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Timestamps
0:00 - Motivating example
3:57 - Deriving the key proportionality property
7:36 - What is e?
8:48 - Natural logs
11:23 - Writing e^ct is a choice

Corrections:
9:40 - I meant to say "*the derivative of* e to the power of some constant..."
12:30 - What's written as "(1 + r)" should really just be r, by any usual convention for how to write an interest rate.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: ang
Vietnamese: @ngvutuan2811

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Various social media stuffs:

[humbleevidenceaccepter]

e is the grade I nearly got in calculus.

[ayushagarwal]

3b1b's videos are like detective novels. First there is a mystery, and then the story slowly moves towards the solution by finding clues.

[deepakachu]

Every child would fall in love with learning if the world had more people like you ♥️

[joejitsuway]

It's amazing the amount of intuition and conceptual learning that is left out of higher education math classes. These videos are pure gold for that.

[arongil]

Your videos never fail to inspire. Calm yet profound, penetrating insights stick out like jewels studded into a ring. Thank you, Grant, for all the effort that you have poured into this series; As a Patreon supporter, I know first-hand the thought that went to these videos on Calculus. I and each member of the audience of 3blue1brown can tell you that your work is appreciated. For me, this means that you were and are at the heart of the reason that I love mathematics: you showed me the beauty. And, as for the rest of the audience, one needs to look no further than the thankfulness of my fellow commenters. 3blue1brown, you are awe-inspiring. Thank you.

[suflaj]

Can we just appreciate the time it takes to animate this numbers, graphs and characters ...

[ristekarashabanov]

After 6 years in electrical engineering, I finally understood the meaning of e, thanks to this video.

[seanryniker]

Like I always say, mathematics is like a well Euler'd machine.

[vidsangre]

Thanks. It's been so long since I studied all this that it is a revelation to me again. All the effort poured into making these videos is highly appreciated by me and all your viewers, I'm sure!

[Squirrel_]

The Internet is such a mixed blessing. There’s the vile poison that is 99% of Facebook and Twitter, but also the treasure that many truly gifted teachers bring us on You Tube. I don’t know which will ultimately triumph.

[jmadratz]

I have a Ph.D. in Aeronautical Engineering with a focus in spacecraft and missile dynamics and control, and applied mathematics and your video never fails to teach me more about mathematical properties and relationships that I never really gave a second thought about.

[AustinWLee]

It will be 10 years since I first learned differentiation..
But it is today that I realized the greatness of 'e'. I am late, but happy to feel the beauty of mathematics thanks to you.
Furthermore, If my juniors can access your channel in the new semester, I would like nothing more. appreciate and have a great weekend!

[ull]

Oh My God ! I could not help myself from pausing the video somewhere less than 10 minutes into the video, and type this comment. My heart was racing because of all that wonderful and beautiful insights I got from this video. Definitely not for the faint hearts !! Your videos are AMAZING !!! I wish there was some award for the best educational video of the year. If then, you would be the winner for sure !!!!

[bluebrown]

The wonderful thing about exponentials is just how many different perspectives you can take to define or introduce them. This video is of course just one look at a possible exploration that could lead you to stumble onto e^x, and if you'd like another, I'd highly recommend Mathologer's coverage of the topic in these two videos:


(By the way, not that it really matters, but the investing example should probably have been written as dM/dt = rM(t), with solution e^{rt}. Although, given the rate at which those dollar signs were coming up, maybe the implication that the interest rate is over 100% was apt!)

[MichelCarroll]

After years of doing calculus, this video finally made "e" click for me. So stupidly simple, for something so anxiety provoking. Thank you!

[BlackRockExecutive]

The amount of work put behind this is simply impeccable, I can feel how careful each development in the story is planned out to tailors to everyone's intuition and understanding. This is a prime example the word 'universal', and should be included in textbooks as visual aids.

[AssasinMilo]

Everyone, everyone, I just got calculus right just now, it finally clicked for me.. if the rate of change over a function is constant across the function the derivative is a constant but if it changes based on where you are in a function that's okay no big deal we just have to represent it with another function based on where you are. The derivative finally clicked for me I was so happy I was shouting out the window telling the world I understand calculus.

[hamiltonianpathondodecahed]

Easy way to remember digits of *e*
After 2.7 repeats a string of 1828 two times and after it follows the angle of a right isosceles triangle i.e45 90 45
e=2.7 1828 1828 45 90 45

[tonycadena]

If you aren't amazed or you don't get goose bumps or even smile in astonishment then your're not really getting what he's saying. Absolutely powerful!!!

[thetntsheep]

The only magic close to the beauty of mathematics is Grant's incredible ability to describe it to laymans such as me