Calculus 6.08g - An Intuitive Proof of l`Hopital`s Rule

[@mousaalsaeed9410]

most intuitive explaining I've ever seen 😲

[@serhatcoban6797]

Then, you dont know 3 blue 1 brown😁

[@mousaalsaeed9410]

@@serhatcoban6797 That's a long time ago

[@serhatcoban6797]

@@mousaalsaeed9410 😂

[@greenhat7618]

@@serhatcoban6797 maybe I’m dumb but I find this explanation on lhopital more intuitive than the 3b1b vid. Ofc his vids are great i loved his vid on taylor series.

[@Miguel_anitez]

Learned about this rule in my Ap Calc AB class a few months ago. I asked the teacher where it came from and why it works and she said to just use it and she didn’t fully know. Thanks for teaching me! I find calculus very fascinating to learn and it’s people like you who make the world a better place. Consider becoming a teacher, you are very skilled!

[@willpugh-calotte2199]

What a great explanation! Sometimes a graphical demonstration like this, although not a rigorous proof to the academic standard, is sufficient for the student to "see" why some rule works. I had a similar problem when shown Descartes' Rule of Signs, but with no proof offered. When I tried to find a proof of Descartes' Rule, the offerings were too advanced for me. Eventually, I reasoned out a graphical justification for Descartes' Rule myself, and that was sufficient to put me "at peace" with the rule and move on.

[@alkankondo89]

This was an excellent intuitive proof! It was carefully explained and easy to understand. So worth the time to watch! Also, your handwriting is very clean and neat!

[@Snillocad143]

I think that the word "intuitive" is an error of application. You are providing a proof explicitly which looks more to have come from first principles and which is an axiomatic reproduction of what l'Hopital would have used himself. Your explanation is clear but there is no circuitous other argument which would avoid obfuscation. For example, for me an intuitive argument would appear to explain l'Hopital's rule without the ideas what you have shown here and thereby move almost to claim credit for the result. Not only would that be a fanciful doubletalk, namely bullshit, but I can't think of how one could connect such disconnected ideas to what we know as an elegant and simple rule.

[@DarkBidhan]

Thanks you help me clear my doubts.

[@hasandogan3510]

Bro you just got that done! Excellent!

[@johnholme783]

Beautiful! Thanks for taking the time to produce this video, everything is now Chrystal clear!

[@pianowhiz23]

I loved this! Certainly one of the better KZbinrs I’ve seen in terms of ability to explain and not leave holes for the student

[@FriedmanArt]

Just wonderful! Many thanks.

[@agrajyadav2951]

Intuitive and awesome!!!

[@IanPryor]

I really like how you graphed each function independently! That is super helpful and offers a new ways to use L'Hopital's Rule

[@gustavotaucei]

Congratulations for the video. Great work!

[@Ed-yk5yh]

This was an unbelievably helpful and clearly explained video, thank you Derek!

[@bp56789]

You know when you know why something is the case, but you can't really explain it, and then someone does it for you so elegantly that your own understanding improves? That's this video. Thanks.

[@DavidPauwelynTravelingbyfoot]

Fantastic. I am sharing this with all my students from now on. Thanks.

[@APaleDot]

Beautiful. Exactly the kind of proof I was looking for.

[@TheEpicGod111]

Fantastic graph and method of explanation. This should be put into all textbooks

[@hqs9585]

Great video!

[@SatishNatarajan]

Thanks for the intuitive explanation

[@michaelmurdoch]

Awsome! Thanks so much!!

[@firstofallbasics1835]

Well explained

[@minhokim8263]

Wonderful, but could you please provide me with more explanation for the case of infinite/infinite? It sounds like limit f(x)/g(x) = limit (1/g(x))/(1/f(x))=d(1/g(x))/d(1/f(x)), which is not equal to df(x)/dg(x)

[@amiyancandol4499]

After that (g'/f')(f²/g²)=f/g , so 2possibilities, either f/g=0 or f'/g'=f/g so there's that

[@KarlBonner1982]

I figured this proof out on my own - and I didn't even remember that it was actually L'Hopital's rule!

[@derekowens]

Excellent! You are thinking the same thoughts as some of the greatest mathematicians!

[@MrCigarro50]

What a fantastic explanation. Thank you very, very, much.

[@fredrikrenstroem1661]

Starting this video: "how the hell does l'hospital actually work?" Exiting: "yeah, that makes sense." Thanks🎉

[@sergiolucas38]

Great video, hope you get more views :)

[@thomasni123]

Wonderful proof dereck. you may want to use Desmos to do your graphing in the future though, it looks a lot neater than the emulated calculator :D

[@tonksonk951]

Hey, I finally understand Lhopital's rule! This tutorial rocks

[@kathieharine5982]

Excellent

[@gregorywong7992]

Excellent explanation yet so simple to understand!

[@m0rtale195]

omg this is the only tutorial that showed why assumption of f(a) = 0 is a valid assumption, thank you!

[@derekowens]

You get it, man! The function has to equal zero at that point or it's not valid.

[@lwendt1]

Excellent explanation. Thank you!

[@locutus7855]

This was beautiful! Thank you. You just got a new subscriber buddy!

[@tranthaptinh]

ONE OF THE MOST FAVOUR MY THEOREM OF CALCULUS ................

[@jeanlemoignan3130]

EXCELLENT explanation, thanks!

[@loopz_attack1658]

i am starting to understand what df/dx means in my calc 1 class. It really helps visualize what i am solving for instead of just plugging in numbers and getting the answer. I feel closer to math every time i see a good explanation of a concept.

[@xxMpEGxx]

you are calculating an infinitesimal difference in height divided by an infinitesimal difference in width = slope same goes for Integrals: that big integral sign stands for Summation and the term f(x) inside is the height times the infinitesimally small width dx which results in the infinitesimal area, which is then summed up for each infinitesimally small dx over the interval

[@quest4knowledge-xh3eu]

the good news is that dx and dy , df etc are NOT infinitesimals. But that's a new discovery. Seek and you shall find.

[@adnansakeeb739]

this was so good....

[@Rocky-me5cw]

That's the best L'H♡SPITAL video.Even KHAN ACADEMY CAN'T beat IT.

[@KarenWasherGrudzien]

Yes, except for the very unnecessary part where he refers to some sort of "curvature"in the earth that does not exist in our reality. The earth is flat, we aint no globe!!!

[@AhmedMahmoud-tv9vw]

@@KarenWasherGrudzien OK Karen

[@KarenWasherGrudzien]

OK Muhammad

[@mathlegendno12]

@@KarenWasherGrudzien Were you joking

[@KarenWasherGrudzien]

@@mathlegendno12 No

[@hansgulbranson9506]

Bless you Derek Owens

[@Rogeriopris]

Of all the explainations that i saw,it was the best

[@howareyoudoing3012]

Very smart explanation, thank you!

[@siyabongankosi9956]

beautiful...plain beautiful

[@akshayv2849]

Smart thinking. Never really looked at it this way :D

[@alphamega3306]

Excellent video! 10/10

[@triton62674]

WOW great intuitive video!!!

[@Lod531]

Brilliant video!

[@sevarchy]

My understanding of this is that limx-->c f(x)/g(x), is "rise_f over rise_g". While the ratio of derivatives is " (rise_f/run)/(rise_g/run) = rise_f over rise_g".

[@fraserpye9667]

thanks

[@allanhenriques2694]

you could also explain it as: if you find the derivative of f and the derivative of g using the definition of the derivative, then you divide them, you will be left with f(a+h)/g(a+h) where 'a' is the value that x is approaching and h is approaching 0. So essentially you are dividing a point on f thats infinitely close to 'a', by a point on g thats infinitely close to 'a'

[@minuklee6735]

it's awesome. thank you.

[@MarioVladutTino]

I know in practice is not useful, but does this mean that when you are doing a limit of the divsion of two polynomials and there is the same numerator and denominator, you can still apply hopital even though is not indetermination? I repeat, I know this is not practical since the straight away answer would be one, I am just saying that Hopital could be applied too. I say this beacause what you explained in the video happens will always happen as long as the functions cross paths

[@GSHAPIROY]

13:15 Not the case, because the derivative of 1/g(x) is not 1/g'(x).

[@jamalsinger9445]

Very very helpful 😍😍 thank you

[@darkdelphin834]

So technically earth is indeed flat if you zoom in by a lot

[@derekowens]

Exactly! If you ignore the terrain. The earth is curved, but locally it is flat. And if it's not flat enough for you, just move in closer and zoom in a little more.

[@zachrox9]

thank you kind sir

[@zaidrehman21]

good explanation

[@TheJunken2]

Amazing!

[@bakdiabderrahmane8009]

3blue1brown style of intuitive demonstration

[@Stejarulpufos70]

What if x -> infinity ? How would this type of graphical proof look like ? I ask this because I've tried every which way and i find it impossible ( my case is a 0/0 case as x -> ∞ where f(x)=1/(4x^2 -5x) and g(x)=1/(1-3x^2) ). Thank you for the video!

[@DoofEvil]

be explains it in the end, where he shows how ♾️/♾️ = 0/0

[@AhmedAdel-ij5bb]

بحسها بتحكي عني انا واختي واخويا بعيد عننا 😭😭 انا شاء الله سيجمع شملنا ❤

[@p.d5545]

Thank you

[@roaahedaya1779]

amazing 👌😍😍

[@Kokurorokuko]

I wonder why L'hopital's rule doesn't work in an arbitrary point where y! = 0. The graphs would be the same but shifted, so it seems to me that the rule should be also applicable. Why is that false?

[@catharsis7629]

Remember these graphs are not straight, the slope would be different in that reference frame.

[@APaleDot]

Remember, we are taking the limit as x -> c, which means we can approximate the functions as if they are linear functions f(c) + f'(c)dx and g(c) + g'(c)dx as shown in the video. It works out so nicely in the video because f(c) = g(c) = 0, which means the ratio between these two approximations becomes (0 + f'(c)dx) / (0 + g'(c)dx) = f'(c)dx / g'(c)dx = f'(c) / g'(c). But consider the case where f(c) = g(c) ≠ 0. Then our limit turns out very differently because as x -> c, dx -> 0 and the ratio of the approximations becomes (f(c) + 0) / (g(c) + 0) = f(c) / g(c)

[@pablote325]

You are amazing

[@miacapaldi822]

I'm someone who has dyscalculia and honestly it blows my mind when this makes sense to me more than simple Maths???

[@loopingdope]

What is dyscalculia? Similiar to dyslexia but with math and numbers?

[@miacapaldi822]

loopingdope yeah pretty much. reading numbers is difficult, as well as counting, remembering facts, misunderstanding place value and anxiety surrounding the subject itself. It's really difficult

[@loopingdope]

You can understand maths from videos like these ones? And this is a rather "hard" subject compared to regular math. It's kind of strange, but best of luck to you and I hope that you'll end up fine.

[@jitendra_9973]

I'm an UG B.tech 1st year student. I've already realized this proof during studying the Thomas's calculus book. And many other proofs like this also have I realized by myself. I think I am a different. What is the best thing I should do ? Because I am confused what I should do. Can you tell me what is best for me ? (Currently I am an electrical engineering UG student) All my classmates are blindly learning coding just to get good job in companies like Google. Please help me !!!

[@johnhugon67]

@@jitendra_9973 "All my classmates are blindly learning coding" Listening to what a random person on the internet tells you to isn't the way to figure out what to do for the rest of your life and is just as blind as what you criticize your classmates for. Understanding a proof like this doesn't make you special in any way either. Do what you enjoy, and if there's nothing you enjoy noticeably more than other subjects, pick something you would be able to tolerate for a long time and pays decently (nothing you couldnt imagine yourself doing for 20yrs or longer). You can search for your passion while you work at a well paying job, but make sure you don't attempt to make anything you dislike your career, you'll hate your life.

[@grjesus9979]

So basically you`re deriving a function respect to other?

[@paulg444]

its might be easier just to use a Taylor series about x=c and L'Hospital falls off the page.

[@tranthaptinh]

PROBLEM WE DON'T HAVE TIME TO DERIVE FORMULA WHEN YOU DO THE EXAMS ........... THE EXAM QUESTION ONLY GIVE YOU ENOUGH TIME TO SOLVE EACH PROBLEM ............... LIMIT TIME FOR EACH QUESTION ................ IF YOU ARE LATE FOR ONE QUESTION THEN THE NEXT QUESTION YOU HAVE TO DO FASTER ...........................................................

[@bhujiamonster3471]

u should use desmos

[@tranthaptinh]

THE PROBLEM ME AND MY FRIEND MAKE AN ARGUMENTS WHICH WAY LEARN QUICKER ...... EVERYTIME GO TO EXAMS ....... I LOST AT LEAST 5 MINUTES TO WRITE DOWN ALL THE MAIN IMPORTANT FORMULA BECAUSE SOMETIMES I FORGET FORGET FORMULA WHEN THE QUESTION IS TOO EASY TO SOLVE ..... REALLY ANNOY ...............

[@tranthaptinh]

ONE OF MY FRIEND SAID TO ME ...... IF YOU FORGET FORMULA YOU HAVE TO LEARN HOW TO DERIVE THE FORMULA HOW TO PROVE THEOREM IN A QUICK WAY ...........

[@derekowens]

Yes! And this one makes so much sense if you understand that little diagram of the two functions and their slopes.

[@dadinggo]

cool

[@vynneve]

bad proof. that's wayyyy too "hand wavy". "dividing" by the infinitesimal dx to get the derivatives in there is just terrible form for a proof. technically does work (if you took some extra steps within that) but there is a much better way, just start with the derivative side, and apply the definition of derivative. (wont write various limits, should be implied) f'(x)/g'(x) = (f(x+h) - f(x))/(g(x+h) - g(x)) (the two denominator h cancel out) We can now evaluate the second terms at a (the lim to be calculated, h is also going to zero remember). But since we are assuming form of 0/0, f(a)=g(a)=0 so, = (f(x+h) - f(a))/(g(x+h) - g(a)) = f(x+h)/g(x+h) = we can now put 0 for h = lim x->a f(x)/g(x) as expected. Much easier imo, and objectively less messy of a proof.

[@derekowens]

It is worth noting how rapidly these ideas took hold. The writings of Newton and Leibniz were adopted by the academic community almost in real time as they were developed. Newton was still alive and had not yet even moved to London when Bernoulli was writing about this rule, less than a decade after the publication of the Principia. And all of this was well over century before Cauchy began to formalize the theory of limits, and a full century and a half before Weierstrass. The point here is that a formal proof involving limits is actually unnecessary. History actually demonstrates this, as neither Newton nor Leibniz, nor anyone else at that time, used limits. Yet they *understood* these ideas and what these infinitesimal quantities represented. And if one understands this, then one understands when one can or cannot divide by dx. It was a conscious choice to approach the theorem this way, without the use of limits. The deliberate aim here is an *intuitive* proof, which I believe can make the concepts more clear and the understanding deeper. For another example, consider what Leibniz wrote to Wallace: "It is useful to consider quantities infinitely small such that when their ratio is sought, they may not be considered zero, but which are rejected as often as they occur with quantities incomparably greater. Thus if we have x + dx, then dx is rejected. Similarly we cannot have x dx and dx dx standing together as x dx is incomparably greater than dx dx. Hence if we are to differentiate uv, we write d(uv) = (u+dv)(v+dv) - uv = uv + vdu + udv + du dv - uv = v du + u dv" Where did the du dv go in this "proof"? If you understand the nature of these infinitesimal quantities, it makes complete sense. And then you can divide by dx on both sides and you have what we call the Product Rule.

[@khalidelgazzar]

Hmmmm .. interesting

[@vanessamichaels9512]

the way you say "over" is so bizarre. Are you like an oldschool so-cal surfer?

[@derekowens]

Haha! That's pretty funny. I grew up on the east coast, and while I didn't do much surfing I did watch Fast Times at Ridgemont High...

[@KarenWasherGrudzien]

2:40 The world is flat!! Get it right n do ur own research we aint no globe! 🙄

[@derekowens]

The earth is flat ...if you zoom in enough!

[@kanerashiki1046]

@Derek Owens What I've learned from this video: zero/zero = infinity/infinity Just kidding amazing video which gave me a lot of insight Now you've just made me interested in the relationship I mentioned above I don't feel like the mathematical community is very comfortable with it

[@kanerashiki1046]

Nor that I can learn much about it from existing resources

[@cpotisch]

They’re both indeterminate forms, but they can’t really be equated.

[@jitendra_9973]

I'm an UG B.tech 1st year student. I've already realized this proof during studying the Thomas's calculus book. And many other proofs like this also I have realized by myself. I think I am a different student. What is the best thing I should do? Because I am confused about my career. Can you tell me what is best for me ? (Currently I am an electrical engineering UG student)

[@REL1C]

Getting off KZbin and finding a passion would be a good start.

[@rekhakalsekar8517]

Flat earth society disagrees

[@davidseed2939]

Very instructive but please use the correct word. A demonstration or explanation but not a proof.

[@derekowens]

Okay, I see your point, and thanks. I do think, though, that the approach shown here is logically compelling, and in that sense it would qualify as a proof.

[@J.B24]

Use Desmos!

[@carsonholloway]

Um no ur actually wrong the earth is indeed flat.

[@derekowens]

Ah, that's good to know. Thanks for clearing that up for everyone. I'll tell my friend who is a pilot and I'm sure that will help him navigate over the north pole, if there even is such a thing. Oh, the silly delusions we would all believe if it weren't for the enlightenment provided here. I'm glad you've shared your wisdom for all of us to partake of.

[@carsonholloway]

@@derekowens I know, it sounds ridiculous at first, but believe me, I used to be as unaware as yourself. But one day, (a rainy Tuesday, to be specific), I had a realization that the government's notion of a spherical earth, and the moon landing, at that, are all dogmatic statements forced upon all the innocent, yet ignorant, members of society. Consider this for yourself; is it really worth the government to spend ~$21.5 billion /year on NASA, for them to make rockets and satellites, or for the supreme leaders of the world (including, but not limited to the members of the Illuminati) to have a spare ~$21 billion /year to continue and expand their operations (the other .5 billion /year is NASA's budget to fake the spherical earth "evidence"). Now, the first counterargument you may give is gravity forces the earth to collapse into itself and form a sphere. While it is true that gravity does exist, and it does keep the world from falling into pieces, (and this is the part where many of the "spherical" earth simpletons seem to get it wrong), a sphere is not the only way for a mass to be in equilibrium. If you have attended physics in modern day university, you will of course be taught that the earth is a sphere as the gravity pulls all the mass into some sort of stable structure, which is only sometimes a sphere; there are many other three dimensional shapes that can be in equilibrium, such as tori or discs. But, as you would have been aware if you weren't a believer of the false information fed to you by the authorities, the earth is obviously a disc. You may be familiar with Ockham's Razor, (originally "Entia non sunt multiplicanda praeter necessitatem", or commonly translated to "The simplest explanation is usually the best" or more literally "More things should not be used than are necessary"), and indeed, in this instance the simplest explanation is that the earth is a disc. The earth looks flat to you as you walk around, you don't notice the curvature, which leads to the much simpler explanation of the earth actually being flat. But instead the government said the earth was spherical, with little to no evidence, and everybody else believed them. I'm genuinely sorry for you, that you're living under a metaphorical rock, unbeknownst to the truth.

[@carsonholloway]

@@davidbandy6268 I'm not a flat-earther, I was just trying to give the most ridiculous sounding argument I could think of.

[@davidbandy6268]

@@carsonholloway I guess I was wrong, it's really sad how much of the world is existing totally oblivious to the lies of our government... And then people like you come along, get so close to the truth, and then just make fun of it because you are too weak to admit that you have been deceived. Unbelievable.

[@carsonholloway]

@@davidbandy6268 Give me your best argument for why the earth is flat and maybe you can change my mind.

[@tunasayl2928]

Thank you