New calc video! Implicit differentiation, what's going on here?

This video took a while for me to fully understand. But, once I understood what you were communicating, I understood the profound nature of implicit differentiation. Great video.

Royal with Cheese

Hmm, you are right that it might be a bit confusing. What I mean by "it's an approximation" is that the actual change to x^2 is not 2x*dx, but 2x*dx + (dx)^2, and likewise for y. So the actual equation to stay on the circle would be 2x*dx + (dx)^2 + 2y*dy + (dy)^2 = 0. But for small enough dx and dy, those (dx)^2 and (dy)^2 terms are negligible, so we ignore them. The effect of ignoring those terms is that you're basically treating the curve as locally linear. A step that satisfies the full expression 2x*dx + (dx)^2 + 2y*dy + (dy)^2 = 0. would stay precisely on the circle, but a step that merely satisfies 2x*dx + 2y*dy will actually stay on the linear approximation of the circle around your starting point, which is the tangent line. Maybe that's more confusing, and doing it in writing certainly isn't optimal, but does that kind of make sense? In either case, I'll take what you said and think about phrasing things differently in the video.

3blue1brown

Hey Grant, thanks for the great video again. I'm a bit confused, and I think a viewer may be, as to what you mean at 10:40 - where you're saying that this is true for the tangent line and not the circle itself. You explained that dS must be zero in order to keep us on the circle. I think what you mean (and please correct me if I'm wrong), is for a fixed value of X and Y, this equation 2x1(dx) + 2y1(dy)=0 keep us on the tangent line of the circle. But if you allow x and y to vary as each point on the circle, from my understanding then it's not actually talking about the tangent line but the circle itself. What is your opinion on this? It's just a bit confusing because yes, it's an approximation but that's when we are holding x and y to some initial, fixed value, not when we're constantly changing it as we move around the circle.

Josh B.

Thank Lionel. Good catch on (a). For (b), you're totally right, a different way to understand implicit differentiation could be to always imagine walking along the curve itself as some notion of time passes. I decided to present things in the way I did in part to emphasize that you can interpret those differentials dx and dy reasonably on their own, hopefully reducing reliance people might have of thinking of derivatives in terms of rates. I think this might better set the stage for things like the role of dx in an integral, or in multivariable calc contexts. I figure good points like the one you make are optimal for things like comment discussions, but there are too many potentially interesting notes and alternate views to always be able to fit into the videos themselves.

3blue1brown

Thanks for the feedback, Magnesium. Good point on the line-length labels. What do you think of making them arrows? Likewise, I like your other points, I'll have to think about them a bit more.

3blue1brown

The ladder analogy and explaining the derivative equality as a constraint we impose rather than a consequence of the defining equation is a nice way to go about it. Implicit is definitely where things get abstract. @ 8.56, you have a close-up in the bottom right where you label the point after movement S(3+dx,4+dy). From the diagram, the nudges are both negative, and you've labelled the lines of nudges to be dx and dy. Usually a variable assigned to a line segment without directionality is understood to be it's length; so people may think that the variables dx and dy are positive in value and so the new point should actually be S(3-dx,4-dy). Perhaps you could label the new point after you state dx and dy are negative. @12.00, you include '=' when calculating the differentials of both sides. Since you're explaining the differential equation as making two expressions match rather than applying an operation to the source equation, I think it's better if the '=' is left out until you say that you're equating the two sides; showing that the equality part is the last step in getting the equation. Similarly for the y=lx example @13.35 Also, I didn't understand why you included that plot for 1/x in the graph @14.34. I can see that the value of the plot is changing in the same direction as the slope of y=lnx, but not whether the two values were equal from the graph. Also, I got a bit distracted thinking about whether the intersections of 1/x with the tangent line and the y=lnx graph had a significance; especially since the tangent line intersection was moving as you went along the graph.

Magnasium

a) There is a very short picture flip around 4:50. b) The difference between the ladder example and the circle example was said to be the underlying time dependency. Wouldn't it have been possible to *introduce* such a dt, e.g. while wandering around the circle?

Lionel Pöffel

Excellent. I have two points.

Lionel Pöffel

Excellent video! No comments at all, it was very clear and precise, and I definitely learned something new.

Hi Dan, Thanks so much for the feedback. To your point on whether the word "derivative" is about ratios or just about the tiny nudge to the output, I actually originally had a little bit addressing that which I cut out of the script. Hearing you talk about it, though, I think it might be worth putting back in there. The upshot was that with a single variable, you can divide by that tiny input nudge to get a convenient interpretation as a rate/nudge, but with multiple input nudges at play, it's less clear what to divide by. I wasn't really sure whether it was worth alluding to common multivariable calculus constructs to deal with that (partial derivatives, directional derivatives, gradient, etc.). At the very least, I think you're right that a few more words are needed on the use of the word "derivative", which I was admittedly using loosely. Thanks again, -Grant

3blue1brown

I'm finally LEARNING stuff I 'learned' 35+ years ago... thank you :)

Christopher Burke

Hey Grant, awesome video as always, and as always, I really appreciate that it helps not just total beginners but also people who've been taught intro to differential calculus but are wondering things like "what the hell is a differential and why do introductory text books always seem to contradict each other on the subject / seem like they're concealing something?" This time I do have some feedback: 1:36 You say x^2 + y^2 = 5^2 is "not the graph of a function" and then move quickly on, but I wonder whether a lot of viewers will be left thinking: "why don't you just rearrange it to get y in terms of x?", i.e. being naive about roots. Maybe worth more hand holding here so that a larger proportion understand why you are having to do something unusual? Same point for sin(x)y^2 = x later. Now 7:30 onwards, regarding the "derivative" dS of the implicit equation. So, previously in the series, hasn't your definition of "derivative" been basically "the ratio of the tiny change in the output to the tiny change in the input"? But now, e.g. at 8:35 you're putting dS on the screen and calling it a "derivative" of S. In the terminology of the series, isn't this a "tiny change" or "nudge" in S, rather than a "derivative"? I.e. if this is a derivative, then where has the familiar "ratio" notion in derivative gone? So in other words I sort of felt like I was about to have expressions involving differentials explained in a way that was completely non-mysterious, and then I was a bit disappointed as there still seemed to be something mysterious after all! (Is the issue that you consciously decided to avoid the word "differential" and so used the word "derivative" for dS?) I have a sense that this is an area that you are particular aiming to demystify -- is there any way you could apply one more level of clarification/explicitness? Or have I just got confused? From 11:00 onwards, sin(x)y^2 = x, usually when you display a graph, it's possible to wrap one's head round it fairly easily, and when you do the animated sliding of the pink point up and down the curve one would usually be thinking, "right, I get this function". But in this case the relationship seems a bit harder to digest. Maybe some more hand-holding to help viewers understand it? I paused and tried to check I vaguely understood the shapes, but the first point I thought about was (0, 1), and then I started wondering why the whole y-axis wasn't yellow (which is a reasonable question, right?). So basically, even with pausing it was taking some effort to comprehend the relationship.

Dan Davison

Another great video! I've actually never seen implicit differentiation introduced in terms of infinitesimals like this. More commonly, I've seen people say "let's treat each side of this equation as being a function of x, and then differentiate with respect to x". Since the functions are equal, their derivatives are equal. I think I like your approach more because it provides more justification for why we could ultimately differentiate with respect to either y or x; we haven't constrained one variable to be an input or an independent parameter. I also like the idea that the equality constrains the types of "nudges" we consider.

Sohan