New Calc video! "Chain rule, product rule, etc. | Essence of calculus, chapter 4"

Good suggestion, Sohan. It certainly would be nice to mention this point, but there are always many many interesting things that are possible to bring up, and I don't want to lose sight of the central goal being to convey the core of a given topic. That said, a video would be hollow without *any* asides like the one you mention, so it's a careful balance.

3blue1brown

This is such great intuition for the chain rule! I just have a couple small suggestions that might help students form a coherent picture in their minds: 1. As I mentioned in my comment on the previous video, the question of sign pops up again when you're discussing the product rule around 11:30. It's important that students understand the distinction between d(sin(h)) being negative here and the use of -d(1/x) around 12:20 in video 3. 2. I think it'd be good to mention the connection between the product rule and the power rule. You spent a decent amount of time developing the power rule in earlier videos, and I think it's worth mentioning that it's just a special case of the product rule. For whatever reason, I've seen a lot of students who think of those as two separate rules and it blows their minds when I synthesize them. On this note, it may also be worth discussing the grouping of terms in the product rule. You chose to split up x^2 sin(x) as [x^2] [sin(x)], but you could just as well have chosen [x] [x sin(x)]. It may be good to mention that, so students don't think that there is a "correct" way to factor products. This discussion could be combined with the discussion of the product rule: depending on how ambitious you're feeling, it could be a cute little intro to inductive reasoning. I realize that both of these points are tiny, and I don't think the video will suffer greatly if you don't incorporate my suggestions. That's a testament to how well you've made these videos and how well they build intuition. Can't wait to see the next installment!

Sohan

Thanks for the catch! Another verbal flub.

3blue1brown

I think I just misspoke. Maybe there's some sense in which "each layer at a time" kind of makes sense, hard to say.

3blue1brown

When you say "mnemonic" around 8:00, it sounds a lot like "noo"-monic (pneumonic?) as opposed to "nuh"monic, which is how I'd expect it to be pronounced.

Benjamin Grossmann

About that opening quote: is "you have to deal with each layer at a time" a correct phrasing? I'd say "you have to deal with each layer one at a time" or "you have to deal with one layer at a time", but maybe I'm getting hung up on nothing.

Benjamin Grossmann

I always find it astounding how often something can be clarified just by doing making that move.

3blue1brown

great video. thinking about multiplication as area is such a great insight

Albert Martinez

Awesome video Grant!

As a maths and physics tutor/educator for the past 15 years I can't believe you've shown me a new way to look at the product and chain rules, particularly the product rule - I've never seen that setup before and it makes it so much simpler! I'm going to get spreading this around at the first opportunity! Thanks for all your hard work and insight :)

Hi Dan, thanks for the kind words! Right now, I'm thinking of just saving some of the topics you mention for an "Essence of multivariable calculus" series down the road. Then again, showing a few parametric things towards the end of this series wouldn't hurt.

3blue1brown

Yeah, that'd be great. What do you have in mind?

3blue1brown

It really is more than just an analogy. The formal definition of a derivative has this fraction at its heart, it's just that you pass it to a limit as dx goes to 0. So as long as any reasoning you do with an interpretation of dx and df as finitely small little nudges doesn't change as dx approaches 0, reasoning this way will not lead you astray.

3blue1brown

Hi Alexander. Thanks for both suggestions. I'm not sure of a great way to go through the full cycle of sine while also keeping x^2 entirely on the screen; what with (2pi)^2 being such a big number, but I'll play around with it.

3blue1brown

Hi Mads. Thanks for the feedback. Just to clarify, which aspect is it that you think warrants more attention: is it the expansion of d(x^2) to 2xdx, or the origin of the full "left d right" term itself?

3blue1brown

Hi Grant. As always, brilliant and insigtful. I have just this single comment. At first, I didn't completely follow the derivation between 7.30 and 7.40. You talk about a little later, with the "left d-right .." rule, but I think a small change in order would make it easier to grasp right off. Also, it's amazing to see that the list of on-screen credit supportes is getting so long that you're soon running out of space :)

Excellent work as always, Grant! Some feedback: 1) at 5:32ish, it might be nice to keep the full rectangle on screen and to show a full cycle of the sin function to really sync the two visuals up in the viewer's mind; 2) at 8:08, you mispronounce mnemonic as 'pneumonic'.

Overall excellent as always, Grant! Since this is a draft, just a couple of things:

I'm from the the US, and same here -- Limits was the first topic we covered. But, I did learn Calc in high school, so maybe it's different for a university class.

Luop90

Hey Grant! Fantastic video as usual. I'm a year 12 student completing my final year of school in Australia. Excuse my ignorance but out of curiosity, why is limits so far down on the list? For our course, it is one of the first topics we covered in calculus. Perhaps this is different across the globe! Thanks, Nad

Is it correct to think about differentials as fractions?Chain rule seems to be just that, does this analogy break down at some point?

Kuba Okrzesa

Also are you considering adding references to further great resources in the description of your videos?

Myles Buckley

I like describing the derivitave as the "exchange rate" between a change in x to a change in y, I don't know why that particular phrasing works especially well for me but it does. Anyway if you feel the same then feel free to use it, it might just be me however.

Myles Buckley

Excellent! One of the great things about this series is the way you make it helpful for total beginners but also contain real insight for people that have seen and kind of understood the material once already. Re topics, is it tempting to have some content pointing briefly to the topics over the horizon that the series doesn't cover in depth? For example, showing that the tiny nudges concept extends to the derivative of a vector-valued function. Or, given that EOLA already existed when this series was made, pointing towards some ways in which linear algebra and calculus come together, e.g. linear/quadratic approximations to a surface? Hm, I guess optimization is another topic that comes to mind, and that could just be 1D of course. (I have watched your Khan Academy multivariable calc series btw, it was very helpful for me.) But of course it's hard to know where to stop and the work you've done on the core is amazing and I'm sure will become very well-known for years.

Dan Davison

That sin(x) slider in the chain rule section is slightly to the right of the other two sliders

Timo Bakr

Yup! I'll cover that in a footnote video between chapters 3 and 4.

3blue1brown

At 0:38 you say you previously talked about the derivatives of exponentials. But you did not. Unless I am missing something? I would really love to see an explanation on the derivative of e^x

Connor Alexander McCranie

Glad you enjoyed. I hope this sentiment resonates with others once the series is released.

3blue1brown

Thanks Aaron, great feedback. I fiddle with a couple versions of the 4:10 scene you referenced, so I might revisit other variants that involve more zooming. You're totally right on the d(x^2) vs. (dx)^2 front also, definitely worth a mention.

3blue1brown

Thanks, Jake! I know what you mean on the tutoring front, where the student says "wait, I totally got this before". This is also a point I feel particularly poignantly as someone creating videos on math. There are limits to how much videos can accomplish, and I find it important to keep that in perspective.

3blue1brown

Nice. The opening actually got me thinking in some possibly-new ways about my own research, in particular breaking down operations into the basics of addition, multiplication, and composition. Some bits of feedback: the screen is pretty busy at 4:10. It might be better to remove some clutter as you go, and maybe even blow up the size of the graphs. My imagination had the derivatives on their corresponding curves rather than their stacked counterparts, and so zooming in on this might make that clearer. At 5:30 and 5:50 the yellow box goes way over the edge of the screen which feels a bit awkward. At 7:00 it might confuse people that you have both (dx)^2 and d(x^2) going on at the same time, and so it might be worthwhile to somehow make it clearer that these are distinct concepts. Hope this is constructive and glad to have this sort of early access!

Incredible. The sliders for the chain rule felt so real. You're really nailing aesthetics in this. No complaints about content either. I thought your comments about your "goals" for this series made sense. The note at the end about practice was a good idea too. I tutor a lot these days, and it really doesn't matter how good an explanation I give for something, if they aren't practicing throughout the week. I'd say one of my main challenges - and fields of experiment - is figuring out how to get students (anywhere from elementary to undergrad) to take their learning into their own hands. Sadly, some students often see a good explanation, feel like they get it, and therefore, and think therefore it's time to move on with their lives. When I see those types of students a week later, they're more shocked than I am that they forgot it. (I'm used to it. Call me a slight cynic.) I read the Josh Waitzkin (Art of Learning) book you recommended on one of your previous videos, by the way. It was helpful both to myself and my approach to my students. So thanks for that too. Keep it up!

Jacob Mirra

Thank you! Really looking forward to the rest!

You are my favorite YouTube channel. I could always do the calculations but now I get where they came from. Thanks :D

Matthew Hausmann

This really clicked with me, the area explanation was fantastic!

And it was a lovely explanation!

Mr. IntelliGent

So excited to watch this...

Mr. IntelliGent

Right as I was about to hit the pillow haha. Nice!