New Calc video! Derivatives of exponentials

The video was good, but I always liked the approach of just looking at 100% continuous growth and discovering e that way. Then you look at what it's like to have growth of 50% and so on until you discover e^rt. and then you can see all other numbers raised to some power as e^rt (this is where the natural log will come in) anyway just my 2 cents

Myles Buckley

I very much liked the vid as always. However I have a counter-proposal to the well-known reproduction example. The pain point with using it is (as you mention in the vid) that it only works out fully "exponentially" at fixed time intervals. You try to smoothen it in-between by considering the population mass. But if you look at it in detail you will end up with a function that connects straight lines. Over each fixed interval the gradient remains constant. One practical example where the gradient also changes smoothly is the length of a rod of metal relative to temperature. This even works for negative values whereas things with populations get a bit awkward at time -1 or so. If this example is too far-fetched is up to you. Just a proposal.

Lionel Pöffel

Oh yeah, nice catch. In case someone else wondered like I initially did; they're uneven in height.

I was waiting for the animation where you graph K^x and then slowly increase K from 2 to e, showing how the tangent slope approaches the value of the function itself.

Thanks for the feedback! I feel like an intuitive explanation of this will ultimately just look like an intuitive description of the chain rule for that specific case. Don't get me wrong, it could still be a worthy thing to do, and could make for a nice concrete instance of what chain rule intuition looks like, but I wouldn't necessarily categorize them distinctly.

3blue1brown

Thanks so much for the feedback! I think you're right that this is a main point worth addressing. Originally, I had intended for this to be a quick insert between chapters, just explaining quickly that these exponential derivatives were a thing. In this case, I'll ponder what can be added to help clarify this importance.

3blue1brown

Great idea, and that's a pretty straightforward change to make.

3blue1brown

Good point! I hadn't thought about that.

3blue1brown

Hi Grant, thanks for the wonderful video again. I look forward to watching them as soon as they're released. I'm wondering if you could delve into the idea of why the proportionality constant is related using the natural log, intuitively. I've only learned it through means of the chain rule, and while yes, that is arguably intuitive, it's more mathematical rather than natural. You start to tie the bits in at the end with the idea that with e^x, the proportionality constant is 1, but when it's not e, you need to multiply by the natural log. So it seems almost like a change of base in a way, but I think that would be something really cool to explain, at least using a change of base rather than chain rule. Like maybe could you show why it intuitively makes sense that derivative of (e^lna)^x is (lna)(a^x)? What are your thoughts on this? Thanks again.

Josh B.

My main gripe with this video is it doesn't give any real sense of what's so special about having a proportionality constant of one. It's like it's the answer to a contrived question, so it comes off as a curiosity rather than as an epiphany. You don't relate it back to the population/mass growth (what's special about a population growing at e^t? how does it act differently?), or even to how it would look on the graph (e.g. the graph at 5:52), and instead once you reveal e, you get stuck in algebra mode, which makes the video just like anyone explaining it on a blackboard, and not like a 3blue video. Would love if you could add more visualization and find some question to ask at the beginning that would tie the video together. Thanks for the great videos

At 3:05 you could switch the t = 1 indicator to increase in smaller increments, e.g. in t = 3.1, t = 3.2... etc and include the mass as a number too (M = 8.5, etc)

Minor note to reduce the cognitive load: You switched from "population size" to "creature masses" but the results were the same, so I found myself listening for when you were going to explain why you switched to "masses" for some time (expecting it to come directly after), which was distracting. (Actually listening back you do slip in that "it better reflects the continuity") Everything stays as integer values for a full 2 minutes afterwards, so the expression "creature masses" seems awkward and distracting. Could you just say "For now we'll talk in whole numbers of creatures, but later we'll switch to a smoother function". And switch to creature masses at the 3 minute mark instead of introducing it so early?

Thanks!

3blue1brown

Just a small typo in your introductory quote: "be" should be "been."

Sohan

You raise a valid point that it's not immediately a given that you can raise 2 to a non-whole number power if all you're starting with is the repeated multiplication definition. But most students going into calc know how you extend the definition of exponents to rational numbers, defined so as to preserve the exponential rule. And from there, extending it to the reals is as intuitive as the real numbers themselves are. That is, delicate in a formal treatment, but pretty straightforward from an intuitive angle. So from a pedagogical view, I don't actually think there's a problem talking about raising a number to a non-whole-number power, and to reference how the exponential property still holds in this case. If we're willing to assume that a student has learned about exponents, I actually think referencing a quantity like 2^{0.00001} is incredibly concrete (what is the 100,000th root of 2) in comparison to most discussion of e^x. I'd be curious to hear what others think, though.

3blue1brown

I'd like to think about this some more and share some viable alternatives if I can think of them. Thanks as always for your work, Grant.

Jacob Mirra

This is one of the most interesting pedagogical problems in secondary/undergraduate mathematics - how to define exp and log. I didn't like this explanation, but I'm not sure if there are any I like. To be logically rigorous, the most straightforward ways I'm aware of are to define log as an integral or exp as a power series (Rudin, lol). I think we can both agree the power series is out because there's no motivation. At least with log you can say "how do we extend the power rule to p = -1" which is kind of motivation. You tried to sidestep these approaches, but in doing so lied to your audience: (1) you assumed without justification that it makes sense to take two to the "dt" power, when, as far as the audience should know, it only makes sense to take two to a whole number power. (2) you then further assume that the "exponential law" should hold for this mysterious operation of raising two to non-whole powers. Making assumptions is okay, but to make these two gaping assumptions only to culminate in an algebraic argument... well I am highly skeptical that you've found the optimal way to teach this (though it's still way better than most of the trash I've seen. It's just, I have such high standards for your work.).

Jacob Mirra

If you give a little more emphasis to the fact that the proportionality constant is less than 1 for base 2 and more than 1 for base 3, I think it becomes more natural when you propose the question of whether it is equal to 1 for some base. The way the narrative is organized, the last few examples are all of integers larger than 2, so the viewer might have forgotten it was less than 1 for 2. Also, it might help to give some fractional examples instead of jumping straight to an irrational base.

André Mello

That's the clearest explanation of e that I've seen. And no compound interest.

Doug Fort

Awesome video, as always. FYI, it looks like the parentheses around "1.0000000" are uneven at 7:35. Also, thank you so much for the work you put into these videos. They're incredible.

J. Dmitri Gallow