Essence of calculus chapter 3: Derivative formulas through geometry
Added 2017-02-01 00:31:52 +0000 UTCHello Patrons,
To finish off January, the early version for chapter 3 in the calculus series is ready for you. This involves numerous visuals which I wish I was shown as a student learning calculus. I don't know about you, but my early relationship with taking derivatives was one of some memorization mixed with graphical intuition for the loose shape of things.
As I say in the video, I think it's important to think directly about the "tiny nudges" underlying derivatives in the context of what a particular function actually means, and how that gives a precise grasp on these derivative formulas. Not just for the sake of proof, or visual satisfaction, but also because it sets the stage for using similar reasoning to understand things like the product rule, chain rule, related rates, etc. Even things I won't get to in this series like partial derivative or the calculus of variations can be notably more intuitive if this is the bedrock of intuition that a student has, I believe.
Curious to hear your thoughts,
-Grant
(There is one point around 16:00 where I must have hit something while recording, so there's kind of a bumping sound. I'm waiting to see if there's more that I want to re-record before fixing.)
Comments
Thanks Sohan! I think you are right that signs deserve a little more attention here.
3blue1brown
2017-03-13 01:18:14 +0000 UTCHey Grant! I'm a new patron, so I'm just now taking a look at these videos. I went through the past comments on this video, and I don't think this point has been brought up (but I think it's worth thinking about). I love your intuition for derivatives as thinking about "nudges". However, I think it's worth discussing more carefully the SIGN of those nudges. You don't mention this at all until it pops up in determining the derivative of 1/x, and at that point you provide very little justification for the minus sign that you introduce. When you're "nudging" a function, why do you only consider positive nudges? That is, why do you only consider nudges that involve increasing a function's input rather than decreasing it? One could just as naturally consider decreasing the input and looking at the resulting change in the output, and this symmetry isn't discussed at all. I think a brief discussion of this could illustrate the importance of keeping track of minus signs, particularly when finding the derivative of 1/x (and later for the exercise of finding the derivative of cos(theta)). Let me know what you think! Aside from that one point, I loved this video. The sine example was beautiful, and your animations are great as always. I'm passionate about teaching, and I find your work extremely inspiring. :)
Sohan
2017-03-12 00:05:50 +0000 UTCIt's not as cool as your other videos, to be honest, but very important for people that are new to math and calculus. Good job, great animations as usual, and elegant explanations. Keep it up!
Benjamin BairMoshe
2017-02-14 15:52:02 +0000 UTCWow! The intuition for the derivative of the sine function was mind-blowing. I had never thought of using similar triangles. Keep up the awesome work!!
Sidhant Rastogi
2017-02-11 18:33:01 +0000 UTCAs someone who is now teaching calculus for the (n+1)th time, I really appreciated your opening note on "why are we doing this?", as I find that I sometimes forget that myself. Keep up the great work!
Benjamin Grossmann
2017-02-06 04:15:09 +0000 UTCSmall nit: 9:34: "three thin squares" -> "three thin slabs"
Martin S
2017-02-05 16:00:00 +0000 UTCNot a stupid question at all! The key is to use the fact that the line tangent to the circle is perpendicular to the radius. It's a bit hard to explain in a text medium, but if you draw everything, and start with that fact, it should help to move forward :)
3blue1brown
2017-02-05 03:31:58 +0000 UTCHi, what is the line of reasoning for the triangle congruence in the "sin" case? P. S. Sorry for the stupid question)
Артем Богачев
2017-02-04 23:17:43 +0000 UTCyou have a random white point on the screen around 11:00
Noam Ta Shma
2017-02-04 14:00:38 +0000 UTCThank you very much for these awesome videos you're making. Aside from the content, that's of course wonderful, I just happen to think it's a piece of art! Just seeing how the "pi" creatures follow with their eyes and movements what's happening on the screen is just mesmerizing! Great artist, thanks again!
MedNait
2017-02-04 12:26:14 +0000 UTCI'm not entirely sure either what the right approach is. I just thought I'd bring it to your attention in the off chance that you'd somehow glossed over it.
Joseph Cutler
2017-02-02 22:39:54 +0000 UTCI think he means d(1/x) is negative, so the length used in the geometric computation has to be |d(1/x)| = −d(1/x). Maybe using the absolute values is better? Thus |d(1/x)/dx| = 1/x², and the sign is obviously minus because the function is decreasing everywhere except the discontinuity at 0.
Roman Odaisky
2017-02-02 12:51:15 +0000 UTC> <i>how any sharp vertex ends up transitioning to a smoother curve</i> Did I just catch you not clicking on that link? ;) Because it is a mathematical proof of the opposite: > <i>While uniform and chordal parameterizations can produce self-intersections, centripetal parameterization is the only one that guarantees no self-intersections within curve segments.</i> Your statement is almost true though; there is apparently <i>only one</i> parametrisation for which there is a guarantee that there will be no loops :P What I'm suggesting is trying out globally changing that curve parameter. I predict that for all the mathematical visuals it will look great; if anything, it should fix similar glitches too in other situations too. In fact, the only place I can see this have adverse effects is if you used Catmull-Rom splines for the Pi creatures (instead of Bezier curves), in which case they would come out differently and need some adjustments. What drawing library do you use? Many set the parametrisation to uniform splines (α = 0, see [0]) by default, because they are the oldest, but often that can be customised. Getting centripetal parametrisation should be dead simple; it's just setting α=0.5, and most libraries expose this parameter, it might be labelled as "knot parametrisation" or "curve tightness"[1] something like that. [0] <a href="https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline" rel="nofollow noopener" target="_blank">https://en.wikipedia.org/wiki/Centripetal_Catmull%E2%80%93Rom_spline</a> [1] <a href="http://p5js.org/reference/#/p5/curveTightness" rel="nofollow noopener" target="_blank">http://p5js.org/reference/#/p5/curveTightness</a>
Job van der Zwan
2017-02-02 10:21:15 +0000 UTCThanks for feedback. I was debating over how much/little to talk about that. Truth be told, I'm still unsure what's optimal.
3blue1brown
2017-02-02 07:01:07 +0000 UTCThe explanation of d (1/x) in the 1/x challenge might be confusing to some. In particular you start denoting "-d (1/x)". However if people solve the equation (x+dx)*(1/x-d (1/x))=1 they will arrive at the "solution" d(1/x)/dx=1/(x^2). Excellent otherwise.
Lionel Pöffel
2017-02-02 01:43:32 +0000 UTCMore great work! I especially liked the explanation of the derivative of sin(x). I do have one quick critique about the way you present deriving the power rule. I presume that most people supporting you on patron are familiar with the binomial expansion, but I doubt that every viewer will be. I was only ever taught how to expand up to third degree binomials, and the notion of how to expand higher degrees is foreign to most of my classmates. Especially foreign is the idea of “choosing”, and using that as an explanation for why all of the rest of those terms go to zero might be confusing. This might just be a problem with my math curriculum, and the explanation might not even be a super important part of the video. Just something that was on my mind.
Joseph Cutler
2017-02-02 00:28:58 +0000 UTCGreat point, thanks Majed!
3blue1brown
2017-02-01 23:13:47 +0000 UTCI wonder if in 13:09 when you say dx = new area that can be confusing to some who might think it means the whole new square's area, rather than just the color-coded additional area
Majed Samad
2017-02-01 23:08:15 +0000 UTCOkay, sounds good. To clarify though, I'm never against using specific examples before teaching a general rule. But you're really asking students to make *two* leaps of logic at a time here: (1) you're having them understand that (rate of change of rectangular area) = l dw/dt + w dl/dt for a rectangle of area A = lw; and (2) you're having them use this formula for the *first time* in a setting where in fact dA/dt is known, and one of the functions' derivatives (dw/dt) is unknown and the quantity being solved for. In other words, it seems to me that on a conceptual level you are (1) invoking the product rule, and (2) using the product rule cleverly to solve for the derivative of one of the factors. It's a great idea for teaching the derivatives of 1/x and sqrt(x) and works wonderfully for both of them (and is much more instructive than the algebraic derivations), but I would have thought (1) and (2) should be handled one-at-a-time, not all-at-once. The point is not that I want you to teach the general rule and then apply to specific examples - I tend to agree with you that that is a bad idea. Rather, I think that task (2) does not help students understand task (1), but rather depends on task (1) being already well-understood. Take it or leave it. I greatly respect your pedagogical judgment.
Jacob Mirra
2017-02-01 22:55:29 +0000 UTCThanks for the feedback Jake! For the 1/x and sqrt(x) cases, I was actually thinking seeing more rectangle arguments like these (along with the initial x^2 one) should help set the intuition so that when I do show the product rule it feels particularly natural. But of course, you are right that one could make the choice to use the more powerful result to demonstrate the specific case. I'll think on it, though.
3blue1brown
2017-02-01 21:39:52 +0000 UTCGood point, that's probably worth fixing for cases beyond this video.
3blue1brown
2017-02-01 21:35:17 +0000 UTCHaha, interesting. What's going on there is that it's transforming the bubble into a giant circle beyond the scope of the screen. The loops are an artifact of how any sharp vertex ends up transitioning to a smoother curve. I guess if I wanted to fix that, I could change what the target shape is. In general, I try to make all curves have as universal a format as possible so that it's possible to transform between arbitrary shapes without messy code handling too many one-off cases.
3blue1brown
2017-02-01 21:34:40 +0000 UTCI think you're right, better to let the image sit longer. Thanks.
3blue1brown
2017-02-01 21:28:55 +0000 UTCThanks! Good catch, just a vocal flub.
3blue1brown
2017-02-01 21:28:07 +0000 UTC12:38 In my opinion, you might consider leaving this exercise of computing d/dx[1/x] off until you've covered the product rule. You'll of course cover the product rule by considering the area of a rectangle of length f(x) and height g(x) (unless you have some other clever thing in mind). Then, you can say, by definition, d/dx[x * 1/x] = 0. That is, x * 1/x is constant. Then use "the derivative of a constant is zero". Don't you think this will jive with student intuitions better after they've seen the product rule? Anyway, the product rule is pretty much exactly the idea behind your exercise anyway, but it's somewhat disguised. 13:09 The same comment applies to this example. Students should note that sqrt(x) * sqrt(x) = x, so that the derivative of the product should be 1, and go from there. The easy fix here is, introduce the product rule before. Yet even as I say it, I see your dilemma: you don't want to introduce the product rule until you have more functions to work with ... is that it? 15:53 Brilliant derivation of the derivative of sine. Surprising, how people have elected for so long to use trigonometric identities and lim(sin(theta)/theta). My god, math teachers must drive you insane. Edit: I reflected some more on this. I was thinking, is it possible that Grant is "hiding" the theory behind the assumption that the circle "looks like a line up close," and therefore teaching somewhat dishonestly? I decided, not at all. For a truly rigorous exposition of the calculus of trigonometric functions, one would need to define them. For that you could go down two routes: define them as a power series, or as the components of an arclength parametrization of the circle. Either way, you need extensive theory, such as the term-by-term differentiation of a power series, or don't even get me started on the arclength parametrization! So, no matter what, your approach has to be non-rigorous in a first calculus course. I conclude, whatever approach is most intuitive, is correct. This is not an Analysis course.
Jacob Mirra
2017-02-01 15:41:54 +0000 UTCReally good work, all of it. My only feedback is a visual one, the scaled curly brackets do not look good to me :S. I don't know how hard would be to draw them in the size you need them without scaling but by actually using some formula to get a nice shape.
Julio Bortolon
2017-02-01 14:52:33 +0000 UTCOk, here's probably the nitpickiest of nitpicks: when zooming in to the thought bubble at 13:18, it shows loops when it should have pointy edges! By freaky coincidence, I just looked up Catmull-Rom splines yesterday, and came across this work: <a href="http://www.cemyuksel.com/research/catmullrom_param/" rel="nofollow noopener" target="_blank">http://www.cemyuksel.com/research/catmullrom_param/</a> With that in mind, I'm going to bet those curves were rendered with uniform Catmull-Rom curves. Replace them with *centripetal* Catmull-Rom curves, and that rendering artefact should disappear, and the curves will look just as good as before in every other respect!
Job van der Zwan
2017-02-01 13:12:52 +0000 UTCI know how to derive almost any of the formulae algebraically; and I knew the geometrical method for x2. But I would never have thought of 1/x, sqrt(x), sinx in this way. Thank you for the beautiful demonstration. Also, even though I read the description before watching the video, I never noticed the bumping sound. In fact, I forgot all about it; that's how entranced I was by your video :)
Magnasium
2017-02-01 13:04:09 +0000 UTCI find the cuts around 7:20-7:25 a little bit jarring - if you don't have enough time for a crossfade, maybe it would better if you cropped the cube and the square animation, and show them side by side? More like how you do it around 8:50-8:55. Although I feel like the cube is fading in/out a bit too quickly there too. I know these are nitpicks, but that just shows how excellent these videos already are :)
Job van der Zwan
2017-02-01 13:00:21 +0000 UTCMaybe Grant updated it since your comment, but if you click through to YouTube, you'll see the playlist of "draft only" videos
Job van der Zwan
2017-02-01 12:33:58 +0000 UTCMight be too late to correct this nit, but a few times you say "leaving behind one minus itself" referring to the exponent in the Power Rule. I thought it was confusing to hear it said that way. Isn't it "leaving behind itself minus one" ?
Steve Muench
2017-02-01 08:07:15 +0000 UTCGreat point, thanks Josh.
3blue1brown
2017-02-01 06:15:03 +0000 UTCAwesome work as always, Grant! At 10:00, I suggest maybe explaining that the derivative part is the difference between the nudged output and the original. Or people may be wondering, "why isn't the derivative x^n + nx^n-1"? And then they are confused on what nudged output really is.
Josh B.
2017-02-01 05:42:07 +0000 UTCGood point, that probably deserves a few more words. Thanks!
3blue1brown
2017-02-01 03:32:23 +0000 UTCFirst, I really wish I had these videos when I was learning calculus. Your videos and the scenes you create are awesomel
Colin Williams
2017-02-01 03:27:36 +0000 UTCFirst, I really wish I had these videos when I was learning calculus. Your videos and the scenes you create are awesome. I did find myself trying to work out how exactly the triangles at the end were similar. Perhaps throw a quick note about how you arrive at this as it is the impetus for your big cos(theta) moment. It seems rushed and not as intuitively obvious
Colin Williams
2017-02-01 03:27:36 +0000 UTCI wouldn't have noticed the noise if you hadn't mentioned it.
Doug Fort
2017-02-01 02:17:13 +0000 UTCUnderstanding how the graph of 1/x is created, it feels like it changed my life, but I feel I'm gonna forget this in a few days. I wish I wouldn't forget all the cool visual understanding things you present. I've been reminding myself of the derivative relationship between the area and the circumference of a circle every other day.
A. N.
2017-02-01 02:16:08 +0000 UTCGood catch! Thanks.
3blue1brown
2017-02-01 01:49:26 +0000 UTCgreat video, the sin/cos part was amazing!
Albert Martinez
2017-02-01 01:14:04 +0000 UTC"leaving behind one minus itself"@~10:10 - very minor misspeak I think.
Jake Palmer
2017-02-01 01:09:28 +0000 UTCIt's still in one of his posts: <a href="https://www.patreon.com/posts/essence-of-1-7245242">https://www.patreon.com/posts/essence-of-1-7245242</a>
Miquel G
2017-02-01 00:47:50 +0000 UTCI missed chapter 1 and can't seem to find it. Is it still available?
Vecht
2017-02-01 00:36:19 +0000 UTC
