Implicit differentiation, what's going on here? | Chapter 6, Essence of calculus

Implicit differentiation can feel strange, but thought of the right way it makes a lot of sense.
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Series like this one are funded largely by the community, through Patreon, where supporters get early access as the series is being produced.
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Timestamps
0:00 - Opening circle example
3:08 - Ladder example
7:43 - Implicit differentiation intuition
12:33 - Derivative of ln(x)
14:23 - Outro
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Italian: ang
Vietnamese: ngvutuan2811
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[@orenshacham9836]

How did the mathematician formalize derivatives? He went out on a lim.

[@plazmotech5969]

lmao

[@otesunki]

UGHHHHH!! so good

[@MrFlameRad]

Please never comment again. 😂

[@joshuarowe8410]

badum tss!

[@God-ch8lq]

pun intended

[@venkatapavankumarsannisett2259]

Would love to see a similar essence of courses on Probability and Statistics

[@uddyantripathi8994]

Man

[@parallellinesmeetatinfinity]

Bhai 😕🤑🤑

[@mathislamiaux6065]

Wut 🤣

[@wrc2933]

Considering how expensive college is, and how good 3brown1blue is at explaining math, this is quite literally a good financial decision

[@RoyalYoutube_PRO]

YESSS... I would get on my knees for that

[@blackpenredpen]

Excellent presentation as usual!

[@sarahbell180]

I say same with you!

[@sansamman4619]

blackpenredpen you learned calculus from watching KZbin tutorials, you don't know how to add fractions, but you know laplace transformation

[@leonhardfrommhold8463]

San Samman you can add fractions by multiplying both sides of both fractions by the base of the other fraction. After that both fractions will have the same base and you can add the top part( don’t know the English word) together. An example would be: 3/4 + 2/3 We expand the first one by 3 and the second one by 4. 9/12 + 8/12 Now that the bases are equal we can add the top parts (9+8)/12 =17/12

[@sansamman4619]

Da Real Doenertier i was kidding xd

[@nullanon5716]

Da Real Doenertier it's "numerator / denominator"

[@pastorofmuppets7654]

Take all my money!! This is the most useful and practical math class I've ever been to. This is better than college and coursera.

[@lxathu]

I share your opinion. I dare to state that I had good teachers... but NOW I know that they didn't have the right tools with the right ways to explain this material. Hats off for the approach, for the quality and the efforts behind them.

[@alekseishkurin4590]

I'm on Patreon immediately

[@iabervon]

He can't just take all your money. He can only take a small amount of your money over a small amount of time.

[@amitwaghmare3863]

ha ha i get that reference!

[@kehana2908]

iabervon you need that small amount of time to approach 0

[@TheFrygar]

Holy shit - literally 4 hours ago I was trying to grasp this concept and thought I'd check if this series had an implicit differentiation episode yet. You are a hero.

[@apteropith]

What I always found most annoying was how both calculus classes, teachers, and textbooks I had (one in high school, one in university) would explicitly teach students to _not_ treat dy/dx as an actual ratio but instead as a single symbol ... and then do _this_ shit two months later (in the high-school case, when demonstrating the derivative of y = ln x, without ever bringing up implicit derivatives). It's never fun to have to unlearn bad advice. It took years, and some headaches with theoretical Hamiltonian mechanics to make it stick.

[@maurofoti526]

The fact is that: they are right. dx/dy should not be threated as a ratio, but is a totaly different concept. Likewise, you can't proof anything with "tiny nudges" and small increments, you would need a whole lot of theory and rigourous theorems. However, wether or not these exaplainations (that are not proofs) are to be considered acceptable, depends on your course of study: a math student will not like this series, but as an engineering student I'm more than happy to think of small increments (without forgetting that there is a whole lot of theory behind that I don't need to remember), and I find these videos as an awesome way to understand complex concepts that can (should) be properly formalized.

[@yigitsezer6696]

@@maurofoti526 i heard it was possible to use infinitesimals and prove calculus rigorously that way. Hyper real numbers and something

[@redwoodenjoyer]

Im trying to self learn it before I take the class, If I end up with the integral OF dx with respect to dx, what does that even mean?

[@Mr.Nichan]

@Boundary Theory The actual important thing to remember, though, is that the "d" is not a variable, despite the fact that they write higher order derivatives in a way that totally makes it look as if it is, even down to the algebra of how they are defined. "dy" is at least variable-like, and so are "dx", " d(2x+5)" and "d([insert anything else])", but the "d" is not. I don't think it's really a function, either, though I'm not sure. It's almost more like subscript.

[@Mr.Nichan]

@Boundary Theory I wasn't implying anyone else thought it was. It's just that I have questioned whether it was or not before, and thinking it is would be a continuation of the idea things which are considered just symbols actually being exactly what they look like. That "d" is not what it looks like.

[@3blue1brown]

Next up is limits! With a look at the formal definition of derivatives, the epsilon-delta definition of limits, and L'Hôpital's rule. Full playlist at 3b1b.co/calculus Also, and thanks to some commenters for pointing this out, when the curves for sin(x)y^2 = x are shown, the y-axis itself should also be marked yellow, as the set where x = 0 is also part of the curve.

[@3blue1brown]

The_StarByte, True! For the circle, you could express it as two separate functions of x, but not all implicit curves are so easily separated. You could have something crazy like x^2 + y^2 = tan(y/x).

[@peppybocan]

And I thought that we will use some integral criterion to derive ln(x) but instead of that you jumped right into multivariable calculus ... that's unconventional and kinky at the same time! :D

[@superj1e2z6]

I still remember the first time I encountered epsilon delta. Frustrating times.

[@cohomological46]

Are you going to make some videos about differential forms? V I Arnold in his book Ordinary Differential Equations says, that differential equation is just relation between differential forms.

[@toxicara]

3Blue1Brown : please start making pi creature t shirts. All you followers would buy them!

[@pmm1767]

we just started calculus in our class and your videos are of great help man. down here in India we tend to memorise log properties and calculus instead of understanding the intuition behind it. thanks for making math interesting and as always a great video!

[@amj.composer]

praanav mahadev menon i assume you are in class 11 like me

[@pmm1767]

Aryaman Manish Joshi yes I am. kinematics chapter rn :)

[@alcapone6796]

praanav mahadev menon you can criticize the system afterwards but right now focus on your studies I have just passed 12th and regret not studying much. avoid this at all cost

[@vishwas425]

praanav mahadev menon bro, dont generalise it to india, there are many good teachers all over india really teach the basic stuff (atleast in my case)

[@pmm1767]

Vishwas Dubey ik but still. cbse2is very dry and bookish in its teaching. there are several teachers out there but only a handful ate good as you say.

[@frogstud]

leibniz would be proud

[@kirtidapandya2918]

frogstud ...Reaallyyy..

[@SuperKnowledgeSponge]

what does leibniz have to do with this video? xD

[@chanakyasinha8046]

He was

[@techwithwhiteboard3483]

@@SuperKnowledgeSponge discoverer and inventor of calculus

[@cursebr2937]

He wouldn’t

[@nicholasandrzejkiewicz]

Even if I've learned this, it's fun to watch.

[@loofusdoofus7735]

Why

[@s.a.chord4879]

Coming back to review calc 1 before I get into multivariable next semester. I never fully grasped implicit differentiation and that made conceptualizing things like parametric equations and optimization really difficult. Thank you for this!

[@cobrachaos6282]

I learned to differentiate x's as normal but anytime we differentiated a y it would be dy/dx. I find how I was taught a lot easier and cleaner but your explanations behind the math were amazing. Thank you!

[@Josh-tu9ji]

Yeah I learned it the same way! I was taken aback when I saw the video implicitly differentiate without y being dy/dx but it makes so much more sense now

[@kikkukun]

differentiating y as dy/dx and x 'normally' is the same as shown in the video, but then dividing every term by dx.

[@leonardchung3825]

Personally I find 3b1b’s way a lot more straight forward, if you’re working with many variables. I frequently forgot to put the dy/dx after the y, and I didn’t really get why I did it, until much later. This video would have helped with that so much, but I guess we all have our preferences.

[@chaotickreg7024]

@@kikkukun Thank you! This is the piece of algebra I was missing. Now my textbook makes sense.

[@v0id_d3m0n]

@@kikkukun multiplying* ?

[@IRisingFuryI]

This is my brain on math. I already know that when this series is over, I'll have huge withdrawals.

[@BiophysicalChemist]

IRisingFuryI There's always Wikipedia, aka the Bottomless Abyss.

[@sebster100]

IRisingFuryI even professional mathematicians get lost in Wikipedia sometimes, lol.

[@myreneario7216]

Oh, yes it´s bottomless. I like the kind of ever-branching definition trees you get on wikipedia. Wikipedia, what´s etale cohomology? Well, for any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X... Schemes? Wikipedia, what is a scheme? An affine scheme is a locally ringed space isomorphic to the spectrum of a commutative ring... Wikipedia, what´s a locally ringed space? A locally ringed space is a ringed space such that all stalks are local rings. Ringed space? That´s a topological space together with a sheaf of rings. Topological space? A topology T on X is a subset of the power set of X, so that the empty set and X are in T, and T is closed under finite intersections and infinite unions. Ok. Then let´s return to the ringed space. Wikipedia, what´s a sheaf? Let X be a topological space and let C be a category. A presheaf F on X is a functor with values in C, that sends open sets to objects and inclusions to morphisms. What´s a category? A category consists out of a class of objects and a class of morphisms between those objects. Each object has an identity morphism and there´s an associative composition of morphisms. Ok, then let´s get back to the sheaf. Wikipedia, what´s a functor? Functors are just the structure-preserving functions for categories. You can call them category-homomorphisms if you like. Ok. But wikipedia, you still haven´t explained to me what a sheaf is. You only explained presheaves. A sheaf is a presheaf in the category of sets satisfying the locality and the gluing axiom. Then let´s get back to ringed space. Wikipedia, what´s a ring? Abelian group with an additional multiplication operation that satisfies associativity and distributivity. Ok. Let´s get back to the locally ringed space. Wikipedia what´s a stalk? The stalk of F at x is the limit over all open sets U containing x of F(U). It´s basically a construction capturing the behavior of a sheaf around a given point. I´ll pretend I understood that. Back to ringed space. Wikipedia what´s a local ring? A local ring is a ring with a unique maximal ideal. Maximal ideal? That´s an ideal whose quotient ring is a field. Ideal? Kernel of a ring homomorphism. Kernel? The preimage of {0}. Ring Homomorphism? Structure preserving map for rings. Then back to maximal ideal. What´s a field? A ring, but multiplication is commutative and invertible. Then let´s get back to the scheme. Wikipedia, what does isomorphic mean? It means there´s a bijective homomorphism Bijective? Injective and surjective. Injective? f: A → B injective iff ∀x∈A∀y∈A:f(x)=f(y) →x=y Surjective? f: A → B surjective iff ∀y∈B ∃x∈A:f(x)=y Homomorphism? Structure-preserving map. For example if you have an operation + on your algebraic structure, then a homomorphism f would satisfy f(x+y)=f(x)+f(y). Then let´s go back to the scheme. Wikipedia, what´s a spectrum? The spectrum of a ring is the set of all prime ideals. It is commonly augmented with the Zariski Topology. Prime Ideal? Ideal whose quotient ring is an integral domain Integral domain? Ring without zero divisors. Back to the spectrum. Zariski Topology? And it goes on like that for quite a bit longer.

[@10bsaivardhan.38]

Woah

[@lontongtepungroti2777]

my mind blown when 1/x came out

[@manavsethi2054]

same here bro

[@thefran901]

It's like when the T-Rex shows up to save the day at the end of Jurassic Park.

[@dxg5659]

I just got to that. I was like “OHHH! So that’s how you get 1/x”.

[@JoaoPedro-ni6jk]

it took me a minute not gonna lie

[@diegofernandoromerodiaz2362]

awesome :)

[@pbpbpbpbpbpbpbpbpb]

Thank you, 3Blue1Brown, for keeping my amazement about math growing. Thank you for doing this.

[@user-hx6qy3ev9x]

I will change to a Mathematics Major because of you. You teach math as I have never seen!

[@Th3Muzza]

It's brilliant, about a week ago I felt really unsatisfied using the standard rules of calculus and thought I'd go about learning their origins rigorously! It's extremely happily coincidental that these videos are being published just as I sought this, as they are excellent!

[@caldrago1470]

I would guess that you are already aware, but when you say "learning their origins rigorously," these videos are far from rigorous.

[@Th3Muzza]

Yeah true, I should really say "intuitively". I wish they'd taught all this stuff back in school. I suppose I could already prove the product rule rigorously from logarithmic differentiation, but like it's really cool to see the way he did it in one of the videos. Though the video on the power rule did prompt me to think "Oh shit!" and come up with a proper proof for it with binomial expansion!

[@BiophysicalChemist]

Th3Muzza Ignore all the pseudo intellectuals telling you that learning mathematics riigorously is separate from or superior to seeking an intuitive grasp of the concepts. The symbolic formalism they mistake as "rigour" is little more than an efficient method of organizing and easily manipulating concepts you are able to mentally visualize. By focusing on simple examples you can easily visualize alongside their symbolic representations, you learn to automatically associate one with the other in the future. Without that effortless fluidity of thought between symbolism and the mind's eye, "learning" math would be as empty and unfullfilling as memorizing the Chicago phonebook.

[@caldrago1470]

+Philip I will accept that insult, but please don't view me as someone who holds the beauty of viewing things the way 3Blue1Brown presents as inferior to rigorous proofs. I would rather be recognised as the smartass faking intelligence that I am than viewed as someone who cannot appreciate the value of intuition.

[@BiophysicalChemist]

Caldrago Haha, sorry if I came across a bit harsh. Your comment was actually totally reasonable, and I have strong feeling about the topic probably made me a bit overzealous. The pseudo-intellectual jab wasn't at you or anyone in specific really, just at my general frustration at the lack of substance in modern math and physics education. Watching 3B1B videos just reminds me of what education could and should be centered on, and the fact that there are so few creative visual and interactive resources that could catalyze math and physics education, despite modern technological resources being virtually limitless.

[@seamusgr]

I know you produced this video series 4 years ago, but your entire series on this that I so happened to stumble upon 4 years later is inspiring me to become a math teacher. Thank you

[@iamanniboy]

how's it going? good luck brother!

[@dickbenis220]

this has been so helpful. im heading back into calculus after taking a long break from it, so these videos have been a fantastic refresher. my teachers never really went that far into what dy/dx means, and so i never really got an intuative understanding of that. so thanks a bunch grant.

[@Margothic]

This series has been of enormous help at college. Thanks Plato I know English, otherwise, I would definitely pass throw calculus without really understanding a thing. keep the amazing work.

[@thundrhawk]

I'm not taking calculus yet, but I've always been intrigued by the ideas. This series has made a ignorant curiosity into knowledgeable fascination. Keep up the great work!

[@daewoonkim6153]

I started smiling at the derivation for ln(x) towards the end for some reason. Thanks for this video!

[@katelynns4887]

It’s really crazy how every thought I had, this video addressed. As the video progressed, the question on the tip of my mind was the next thing being discussed. You don’t know how extremely happy I was to find this series as I was never satisfied with my understanding of calculus throughout my entire year. I always wanted to know the “why” behind the things we were just supposed to accept. This video was everything; thank you!!!

[@ToxicJassassin]

7:32 "But for the ladder (latter) question…" not sure if that play on words had already been noticed 😂

[@korayacar1444]

Skillmau5 not to mention that it was addressed as the former question in that specific scene

[@defectus1769]

it wasn't a play on words

[@kavinbharathi]

@@defectus1769 But it was beautiful nonetheless

[@oussamawahbi4976]

I wish there was videos like this when I was studying calculus

[@HerbaMachina]

Right exactly what I thought, especially with my calculus teacher XD

[@alacastersoi8265]

so did he.

[@MATHBOSS287]

you would probably be having a very painful time understanding him as a student, just like me right now. i know he could be the best one to explain this but i just can't see things as smooth as he is trying to show us (its driving me crazy)...... ( its seems like we have got to go through lots and lots of (blindly working with derivatives) and then figure out after a punch of years that this was the beautiful meaning of everything we used to blindly study.

[@PierreThierryKPH]

Will you keep blowing my mind? This is something I loved when I started maths at University level: almost everything that you use, you see the proof of.

[@ralphinoful]

Damn... I'm currently finishing my last semester of undergrad, taking real analysis, and although you're not going very deep, this is giving some incredible intuition.

[@MarcusHCrawford]

Your videos-your compassionate, relaxed but entertaining, multimedia approach to education-is excellent. You set a high bar with the first videos in this series, and you have yet to make one that I felt was lacking any of the passion of its predecessors. You sir, are a gift to open-source free education.

[@phunmaster2000]

when the graph of sin(x)y^2 = x is plotted, why isn't there a vertical line through the origin? when x = 0, the equation would be 0y^2=0, which is true for all y values, yet no such values are plotted.

[@slartibartfast336]

Yep, good catch.

[@3blue1brown]

True! Good catch.

[@davidebellucci5207]

In my opinion, it's not true. If you take into account the limit of x/sinx for x that tends to 0, the limit is equal to 1. So the equation y^2 = x/sinx is true only for the values 1 and -1 when you go around 0.

[@slartibartfast336]

Davide Bellucci None of what you said is relevant. (0,y) satisfies the equation for any y, so the graph of the equation contains all points on the y-axis.

[@davidebellucci5207]

Slarti Bartfast You are wrong. If you isolate the y and take the square root, the result is two functions which cannot have more than one value for each value of x. So you have to accout for the limit.

[@CasualGraph]

I love what you've done with multivariable functions.

[@joshsutton6648]

In my first calc class as a junior in HS. These videos really help. I watch these after learning the lesson/chapter and everything becomes so much more clear.

[@NyikaWachira]

I learned Implicit Differentiation a week ago and we took the derivative with respect to x of both sides, and then we were left with dy/dx, which we solved for to get the derivative. Its very interesting, but I didn't find it too far-fetched from simple differentiation

[@dudeman3981]

So pumped for this next video! Can't wait to see our take on the formal definition of a derivative.

[@Kyle-bg8nu]

My ears have never been blessed with such a soothing voice. You are the essence of a great teacher.

[@tristanc6967]

Showing the similarity between the ladder problem and the equation of a circle was something I have never seen before, and you explained it so crystal clear. My jaw dropped as you calmly went through it all. What a talent you have for teaching and presenting, thank you very much!

[@samt3412]

I didn't realize the circle equation was a model of the ladder problem, despite having seen both a bunch throughout my Calc class, until now either.

[@florianvancitters3674]

As an international student at an English University I'd like to truly thank you for these video series. Many of the concepts of the videos are well known to me, but the logic used to look at them and some methods are not. This video series has been an amazing way to 'review' certain topics I simply did not learn back home.

[@amolven]

HOW have I not found you for a whole semester?

[@dineshydv]

Never thought calculus was so interesting. This series rekindled my interest in calculus. Way to go.

[@iWaZziT]

i love to watch these videos, getting inspired by your ideas and animations and just pause to play around with the equations, thanks for making that possible, keep it up!

[@grantstenger6182]

This is so clear, thank you so much for all your hard work!!

[@corydiehl764]

I'm kind of inspired by this to make a series of videos on the less well known branches of calculus. For example, product calculus and fractional calculus aren't taught at most universities, but have very powerful meanings to their applications. If anyone has any tips for getting this started, let me know.

[@andresmolinavillarino3312]

Jesus Christ... that quick explanation of the derivative of ln(x) was majestic :O

[@CubicOctagonalgaming]

I'm currently revising all this for my exams and this was the one topic I was weak on so thanks a lot :) this helped a bit.

[@Gold161803]

Nice approach to the derivative of ln(x), cool way to apply the video's material. But I've always preferred the variable inversion method: If y=e^x implies dy/dx=y, then x=e^y implies dx/dy=x. Done!

[@anshumanagrawal346]

Great now prove dx/dy = 1/(dy/dx) :)

[@ericbandera9480]

Wow. Been trying to get into diff eq, and realized that I needed to review implicit differentiation. The general mechanics of solving implicit derivatives has previously seemed easy, but I think there are more complicated concepts going on under the covers. You outlined around 8:45 how 2x(dx) +2y(dy) would need to equal 0 for the second point to be on the curve. It makes perfect sense. However, for the early point with which they usually teach implicit diff in a Calculus class, the notion of an infinitesimal on one side of an equation is daunting. So, they typically teach (d/dx)(x^2")+(d/dy)(y^2)*(dy/dx)=(d/dx)C. But this leads to the inevitable - that y is not a function of x, and therefore dy/dx which should only be dependent on only x per the formal definition of derivative, is now dependent on both x and y (because if y is + yield opposing dy/dx from y is -). And therefor breaks everything from a rigorous standpoint. Now I see much better that if you accept 2x(dx)+2y(dy)=0 that dividing all by the specific(dx) will solve the problem with established specific positives and negatives for each dy and dx for the given x and y. I hope this blows someone else's mind too that was having the same trouble as me. Thanks for the awesome videos!!

[@nikhilnagaria2672]

:o I noticed this now thanks

[@erinhopper6568]

you're literally saving my calc experience thank you so much for these

[@user-en7dx1qp3k]

I took 7 months of AP Calculus so far and this is the video that made me truly understand implicit differentiation. Thank you.

[@TheTruthSentMe]

That 1/x derivation blew my mind.

[@RAyLV17]

YESSS! Please make a video on Chaos Theory Love your channel!

[@figurehe4d]

I love these videos because I learn the process and algorithms in class then I come here and watch Grant's explanations and see all the mechanics become beautiful.

[@ayzack2361]

Thanks for all of these videos. I am currently reviewing calculus because I want to fully understand the concepts and you are doing an amazing job.

[@KOPLuffy]

amazing, perfect solidification of my knowledge on implicit differetiation!

[@datamancy138]

nice that you're putting this up right before the AP test. ;^)

[@nikitanugent7165]

These videos are great. I'm transferring back into engineering soon, and I want to actually have a good grasp on the math this time (as opposed to the last three times I learnt it). I really feel like watching these videos before I sit down and crunch a bunch of problems will help me understand what I'm doing, and therefore be better able to remember it.

[@michalchik]

I love the videos this guy does. these are so well done in so many ways on so many levels. I say this as a math and science teacher, a researcher and someone who loves the deeper aspects of how science, math and art connect. Bravo Sir!

[@bumdeedum8771]

When was the last time u were excited for a youtube series? . . . . Ik! During Essence of Linear Algebra ;) btw a+(n-1)d th

[@vishwas425]

BumDeeDum lim x-->0 (1/x)

[@bumdeedum8771]

Vishwas Dubey does not exist ;)

[@CMDRunematti]

never...

[@SKO_PL]

That's the point

[@bumdeedum8771]

SKO oh! i realized it now. XD

[@morethejamesx39]

Normally I have to be in the right mood to watch maths videos but these are so interesting they're on notification now

[@ItsJoeyG]

That explination for the dirivative of ln(x) was astounding. It imeditally just clicked. Amazing work! Thank you so much for this, it's made me apresaite my calculus classes so much more.

[@flwi]

Wow, this is the best explanation of that topic I've seen so far! Thank you for creating such great content.

[@byugrad1024]

"the latter question..." vs. "the ladder question..." 1, 2, 3, GO.

[@dayofdarknes]

Genius use of “with the ladder/latter problem” I commend you

[@zdoumagne]

This is the reason why I wake up each day this week. Grant you are my role model, your essence of linear algebra series was great and helped me excel in my linear algebra course. Your essence of calculus course so far as been so eye-opening even after I finished my calculus courses.

[@havewissmart9602]

u know his name???

[@zdoumagne]

you can check out his voice and look at the first video on multivariate calculus on khan academy, and you can see his name at the end of one of the videos from this series

[@jasonwillows5239]

10:52 I just want to outline what phenomenal teaching this is. Having no knowledge of implicit differentiation outside of this video, I paused the video at this point and tried to solve what the derivative of this formula was, and I got it right. I'm not a smart guy either, it entirely lends to how effortlessly this guy manages to break down such a complicated subject, I can't overstate how impressive this series is.

[@bb3ca201]

J/k :) I just learned more in this cool vid than in nearly an entire year of Calculus

[@danielettedgui148]

A real pleasure to hear such clarity. Thank you.

[@NITURROY]

I'm in calc and I was thinking about what implicit differentiation actually means and what it might look like graphically and no one I asked could clearly explain it to me. I've been searching for that answer for hours and days on end and i eventually found this channel =) Just Wanted To Say Thanks & That You're Awesome

[@roycrippen9617]

You videos make learning calculus at university so much easier, thank you!

[@yy8665]

This literally made me understand math in a completely new way. Props to you man.

[@c64cosmin]

10:20 that moment when it clicks :D, Thank you very much!!!

[@sprotzy7338]

Thank you for explaining this I've been having trouble understanding what dx/dy even is but you explaining what they are has really helped

[@ashrafallislam8027]

thank you so much for these videos you explain calculus in a way that's really fun and captivating whilst also explaining every detail. It's common in maths for students to take the concept in without actually understanding it fully like they may know how to calculate dy/dx but they don't know what it really is. Thanks so much for the video!

[@unclegranpawafiaahmedyahia5925]

Au plaisir de voir next chapter ..

[@d-thec-tieve4648]

12:00 amazing video as always, but i feel adding a 1×dx instead of just writing dx on the right side of equations would take this video a dx further

[@leonardsalt]

The way that equation clicked at 4:36 blew my mind. Please do more advanced series!

[@nihil1]

It would have been better to have watched these videos over and over again throughout the semester instead of having the calculus I classes I had. This is a true jewel of both calculus and teaching.

[@junesept234]

I can't understand some of it, because of my faulty brain. But my heart love it.

[@emlmm88]

You're videos are very well put together. I wonder, will we get a multivariable and Diffeq series?

[@gabrielestes7236]

Absolutely excellent teaching. I’ve never found calculus this fascinating or this well explained

[@dlevi67]

Best video in the series (so far!). Thank you!

[@negoitamihai8251]

It took me a while to take this in, but I have here an explaination which considers simpler examples of graphs, which is also meant for myself as a reminder if I ever doubt anything about implicit differentiation and end up coming back to this video. Please tell me if you didn't understand something from it or if I said something wrong. If you want to nudge both sides of an implicit equation and still be on the graph that the equation defines, the nudges to both sides must be equal to eachother. For example you have an equation y=2x and you take an point (1;2) as an starting point so your equation will look like this: y=2xy=2*1y=22=2*12=2. so your equation ended up 2=2, by replacing y with 2 and x with 1. An different point on the graph, let's take (2;4), gives us the equality 4=2*2, 4=4, representing point (2;4), in fact, every point that belongs to the graph is represented by an true equality(2=2 or 4=4 or 4,0001=4,0001). If you nudge the first point (point (1;2); 2=2) by an dx and dy( difference in x and difference in y) by, let's say, both dx and dy of 1, we end up with y=2x; (y+dy)=2*(x+dx);(2+1)=2*(1+1), we get 3=4, which is false, telling us that after an nudge to x of 1 and y of 1 the equation is no more valid, which, obviously, means we are no longer on an point the graph, we, actually, ended up on the point (2;3), which does not belong to our graph. For dy and dx to keep us on the graph, we must have the condition d(y)=d(2x),meaning difference in y (d(y)), must be equal to the difference in 2x (d(2x)), meaning the change to both left and right expressions must have the same quantity for the equation to remain true(for us to still remain on the graph), for that equality to still represent an point on the graph. By expanding the condition a little bit, we can leave d(y) as it is for now and let's first see what that difference to 2x with respect to x is. Which is 2*dx, obviously, because for every little or big difference to x, the expression 2x will double that. So we get d(y)=2*dx. And remember what our condition was? We needed to find a way for both expressions changes to be equal to eachother for it to still represent an point on the graph. And by rearranging d(y)=2*dx, we can divide by dx and get d(y)/dx=2. Which tells us that for every quantity of dy(difference in y), we need it to be twice as big as dx(dif in x), for the equation to still represent an point on the graph. Let's take our first example, point (1;2), represented by 2=2, y=2x; and our condition said that for every quantity of dy, it must be twice as big as dx, so, for an dx of 2, for example, dy must be 4.(y+dy)=2(x+dx); (y+4)=2(x+2); for our point (1;2): (2+4)=2(1+2), so we get: 6=6, which is an true equality, so we have jumped to another point on the graph with our nudges dy and dx, which is actually (3;6). And of course, in calculus, we consider very small quantities of dx so the point that we will jump on will be very close to our starting point, so that our different point will show an ratio of dy and dx very accurate for that little zone of the graph that we want to analyze, so small in fact, that we can say that it keeps us on the tangent line to an graph at the first point, for every graph, curvy or very curvy, because limits of course. I gave an simple example so you can see what is happening with an straight line, for which you can say that after an jump of any magnitude to another point you are still on the tangent of the graph, or the graph itself, which are the same in the case of a straight line.

[@technoultimategaming2999]

Me before calculus "God, I love this man and this series. It's amazing and really helpful" Me after Calculus II "God, this series is so cool I love watching it for the 100th time!"

[@Answeriz42]

This explained implicit differentiation so much better than any of my previous calculus teachers explained it, bravo sir.

[@isaacdeutsch2538]

I watched these videos last year when I took calculus, because they give brilliant names and illustrations to the intuitions I vaguely had for things in the course. This year I'm taking multivariable, and found the proofs of total differentials and the multivariable chain rule understandable but not at all intuitive. But when 3b1b explained implicit differentiation as dealing with a single multivariate function, and animated dS = 2xdx + 2ydy, I instantly understood with perfect clarity why the total differential and chain rule are defined the way they are. I'm not sure if this intuition remains the same for other multivariable functions, but this was really amazing. I think it would be really cool if 3b1b could do some videos on intuitions behind multivariable calculus ideas, since a lot of them are hard to visualize and conceptualize.

[@neilsucks]

@7:30 "...but for the ladder/latter question..." pun intended, 3B1B?

[@bantonsmith592]

This is a beautiful video but I am a bit confused about dx/dt being 4/3 at 7:16. The ladder is 4 meters above the ground at the top and its base is 3 meters from the wall, the top of the ladder is dropping at 1 m/s so it will travel the 4 meters in 4 seconds. The ladder is only 5 meters long so it will travel 2 meters further away from the wall (it's already 3 meters away from the wall at its base and it is only 5 meters long so it can only travel 2 meters) in the same 4 seconds so dx/dt should be 1/2 m/s. Or am I missing something?

[@edwardfanboy]

You aren't looking at the average speed over a finite interval (e.g. 4 seconds), you are looking at the speed at a given instant in time. At a different instant in time, the speed will be different so you can't extrapolate one speed over a full interval of time.

[@TensegrityEnergy]

You are absolutely correct, now that the ladder slides down the wall in a straight line and thus the movement of the ladder is no longer described bij a circle, hence X^2 + y^2=5 is not a valid description of this movement/problem. However if they would pulled the wall away in a split second and the ladder would have fallen without sliding back on the floor (along the x-axis) then the movement would have been a circle and the formule and dx/dt would have been correct.

[@isavenewspapers8890]

@@TensegrityEnergyThere is still a theoretical circle here, but its center is moving along with the base of the ladder. If we fix the camera to focus on the bottom of the ladder, we will see the top trace a circular arc.

[@socalacura1338]

After all this time, you've finally shown the intuition behind the "related rates ladder problem"! If only I had known this four years ago...

[@ysink]

Differentiating ln(x) just seemed so reasonable and easy to follow even though its a lot at play together - amazing series!

[@zelllers]

7:30 Did you say "But for the ladder question..." or did you say "But for the latter question..."? I have to know!

[@Lucy-ng7cw]

Yammy Hammy subtitles say ladder.

[@iqranthing544]

I was also confused 😂

[@quantised1703]

PLease make more videos on fractals!!

[@dragoncurveenthusiast]

Yes, please!

[@davidcora2751]

Thanks for sharing your knowledge. I’ve had some college calculus and I’m a bit out of practice. Again, thanks for sharing your knowledge.your videos are very informative and interesting.

[@kevinding1204]

OMG THANK YOU SO MUCH!!!! I've been trying to understand this for a while but this video just made it so clear for me!

[@MozartJunior22]

There is a better way: A small movement on the circle is still on the circle, so (x+dx)^2 + (y+dy)^2 = 5^2 therefore x^2+y^2+2xdx+2ydy=5^2 since x^2+y^2=5^2 then 2xdx+2ydy=0 dy/dx=-x/y

[@Uejji]

The algebra approach isn't "better" so much as it is a way to reinforce the concept of implicit differentiation on a simple equation. Since the equation in question is a polynomial, it is trivial to find a derivative without relying on calculus techniques. Much how we can take derivatives of polynomial functions using the difference quotient. It becomes less trivial when applied to transcendental functions. For instance, sin(xy) + e^(3x) = 7y Using implicit differentiation, we can easily reach cos(xy)(x dy + y dx) + 3 e^(3x) dx = 7 dy Expanding: x cos(xy) dy + y cos(xy) dx + 3 e^(3x) dx = 7 dy Rearranging: y cos(xy) dx + 3 e^(3x) dx = 7 dy - x cos(xy) dy Factoring: [y cos (xy) + 3 e^(3x)] dx = [7 - x cos(xy)] dy Dividing: dy/dx = [y cos (xy) + 3 e^(3x)] / [7 - x cos(xy)] This would not be a 6-line problem using algebra.

[@columbus8myhw]

+Uejji Note that we know that sin(x+dx)=sin(x)+cos(x)dx due to derivatives. So it's doable (but annoying).

[@Uejji]

Is that so? Let's check it. Let x = pi/2 Let dx = pi/4 x + dx = 3pi/4 sin(x + dx) = sin(3pi/4) = sqrt(2)/2 sin(x) = sin(pi/2) = 1 cos(x) = cos(pi/2) = 0 sqrt(2)/2 = 1 + 0(pi/4) = 1 So, this is not true.

[@Uejji]

Let's elaborate: sin[(x + dx)(y + dy)] = sin(xy + xdy + ydx + dxdy) Using our angle sum rule for sine from trigonometry ( sin(a+b) = sin(a)cos(b) + cos(a)sin(b) ) we can (through some rigor) come up with the following sin(xy)cos(xdy)cos(ydx)cos(dxdy) - sin(xy)sin(xdy)sin(ydx)cos(dxdy) - sin(xy)sin(xdy)cos(ydx)sin(dxdy) - sin(xy)cos(xdy)sin(ydx)sin(dxdy) + cos(xy)sin(xdy)cos(ydx)cos(dxdy) + cos(xy)cos(xdy)sin(ydx)cos(dxdy) + cos(xy)cos(xdy)cos(ydx)sin(dxdy) - cos(xy)sin(xdy)sin(ydx)sin(dxdy) And that's just the expansion of *one* term. Good luck removing all of the dx and dy from those trig functions in order to group, factor and divide out. Now, maybe this can be done with algebra through some ingenious steps (although I strongly doubt it), the point wasn't that it's impossible, it's that saying that the algebra way is the "better" way is fairly naive and really only applies when examining nicely structured polynomials, which in the world of higher mathematics we cannot limit ourselves to.

[@m9jbhakar]

as dx tends to small values, the equation becomes true.. afterall thats what "d" means in "dx"

[@TheWombatGuru]

this is where I'm starting to have trouble

[@chaotickreg7024]

The idea finally clicked while I was watching this video. Thank you from 2022!

[@baptistemary9385]

Before anything, all the series are excellent. I have a two related suggestions to improve the explanations on this video. First is a proposition of graphical representation to help understand those relations like the one for the circle: a 3D graph ! If you represent f(x,y)=x²+y² in 3D, the relation x²+y²=25 is more clear: this is the intersection of the surface f with the plane of coordinates (x, y, z=25). By the way, and this is my second proposition, this is the very good moment to talk about the unfortunate polysemy of the equal sign. The equal sign in a relation, in a definition, and in an equation gets different meanings. It is important to be aware of that, and I feel it is a good occasion to talk about that.

[@TwentySeventhLetter]

Every time I hear you say "dy," I can only hear Daddy Yankee yelling "DEE WAAAAAH" in _Despacito_

[@namitanene3531]

Great, now that's all I hear ,too!

[@srivatsajoshi4028]

wtf dude! now i cant not hear it. thanks

[@kubbasz9315]

please come to brazil

[@ajcollins15]

Now that I just finished all of calculus, it's interesting and fun to go back and see this whole thing all over again! Like related rates when I was in calc 1 was impossible, but now that I finished calc 4, related rates is way easier in 3D calculus! But I'm loving these videos!

[@aalhadmone6463]

Impeccable presentation, powerful conceptualisation and excellent visualisation - perfect, as always.

[@josephthomas4023]

Great! 29.5/30 :)