L'Hôpital's rule example 1 | Derivative applications | Differential Calculus | Khan Academy

[@BryhanEspinosa]

yo dawg I heard you like limits so we put a limit in a limit so you can solve for the limit by using the limit of the limit for the limit of that limit limit's limit to get the limit.

[@stupidagface1]

+Bryhan Espinosa this explanation is so clear. thanks man

[@narayansati3828]

this explanation exceeds my thinking limit

[@highmars2626]

UNLIMITED!!

[@liamjames4742]

limitless

[@kevinnimalan9351]

Enlightening

[@Houndtrooper]

Limitception

[@midsummerstation3345]

nice one

[@DaelinZeppiTheComputerGamer]

Starring L'Hopital and Sal Khan!!! Coming this summer!

[@Salma-qy3qb]

Best joke ever I spit the tea all over the wall😂😂😂

[@isavenewspapers8890]

i'd say derivativeception, since we take the derivative of a derivative of a derivative

[@Sam-fq1ho]

brilliant

[@MarkToast99]

my prof calls it the insanity rule because you'e literally trying the same thing over and over

[@abhirajarora7631]

I like your professor.

[@theexplorer9927]

You’re profesor is realistic, I like it

[@chisomopara3414]

So do we continue to use l'hopital's rule until we come up with a non-indeterminate answer?

[@midsummerstation3345]

yeah..99.99% cases i just did that and calculated some hard limit problem way more easily

[@fungawi]

yes we can so

[@thedramallama3089]

yea

[@Economic-_-]

In actual you took it once if you consider last one as your function

[@sandy69402]

@@Economic-_- bro u are literally replying to a 7 y/o comment

[@g2k25d94]

You know this guy is something special when he gets the same thing 3 times in a row yet still trudges on!

[@hughrussell2656]

I wish I was this excited about maths

[@merritt2014]

L Hopital's rule, aka that rule you wish you would've known in Calc 1 lmao.

[@grobbyman]

We're learning this in my calc 1 class. Although we've been done working with limits for a while now.

[@shandevin5417]

im in calc 1 right now studying this cus its on out next test

[@notsomortal4972]

Got my test today. Wish me luck guys I'll need it

[@purpleapplepie2276]

Luis Castro I hope u failed

[@alexploughman9123]

@@notsomortal4972 Hey bro did you pass?

[@ah2582]

Sal, your a genius...if only you were my personal teacher

[@DGVFX]

This is great!! I wish their was also an engineering or computer programming section that would be amazing Thumbs up if you agree!

[@kathryniii]

omg! this is so crazy...my calculus final is tomorrow, i hope the test won't give problems like this ..aint got time to take 100 derivatives -__-" anyways, thanks for explaining! the video helps me understand better! ^^

[@bdgrey]

I call shenanigans! Math is beautiful

[@jimmyalderson1639]

How do you know when to give up on L'Hopital's rule? I understand the logic that 'this could've been your starting pointm' but when do you realise the rule won't work?

[@thedramallama3089]

when the conditions aren't met. lim g(x) and lim f(x) need to be zero, and lim f'(x)/g'(x) must be definite

[@thedramallama3089]

probably

[@departmentofanalytics1116]

you take the limit of L'Hopital's rule as the number of times you use it approach infinity

[@vasudhagupta3514]

@@departmentofanalytics1116 I laughed out loud because I thought this as an excellent joke but you're like 'department of analytics' so.....um you were kidding right? Or is this some actual thing I'm not getting...

[@tarannumsk761]

When you get a constant value

[@Subbu1811]

Thanks a well. Example on L'Hopital's rule explained in an excellant manner which faciliated me to comprehend the rule without any ordeals.

[@Ganondorfothraccount]

@Charddy I find it very helpful because it reinforces his previous statements, and becomes more concrete in my mind. That being said, everyone is entitled to their own opinions.

[@arnsassassiner]

i have my finals this weekend. u are my hero !!!

[@Salma-qy3qb]

You're so excited that your videos make me happy❤ we love you ,Sal❤

[@nddung92]

Thanks you very much, my 2nd best teacher (after my parents). Your lectures are always interesting.

[@superdupe8]

haha i can just picture sal going to the 100th derivative and saying "so does the limit not exist? NO! we have to take the derivative AGAIN!!"

[@Frankenberry84]

Awesome video, I have a Calculus I test tomorrow on this and this cleared up any questions I had. Thanks.

[@SkywarsLobby]

how did you do?

[@internationalremixes6440]

thanku so much.....u don't know.........u hv helped me how much...i'm in cbse class 12.and i've just strucken at one of the NCERT exemplar problems of continuity and differentiability but ur this video made me do that....thanks my brother..i love yaa

[@gairick9]

I hope you were aware of the fact that you cannot use that in an CBSE board exam as that is not included in the syllabus

[@swagotosurjodutta7341]

CBSE is bound to give marks *if your answer is correct.*

[@dev3609]

So keep diff till you get a value other than 0

[@amandanguyen2792]

Super helpful! Prepared me for my exam!!

[@maynardtrendle820]

Honestly amazing!😮

[@saburousaitoh]

Please look the paper: [29] viXra:2001.0091 submitted on 2020-01-06 17:52:07, (58 unique-IP downloads) Division by Zero Calculus for Differentiable Functions L'Hôpital's Theorem Versions

[@chimphead73]

Sal Khan you magnificent man you've done it again!

[@mrwansabah]

in the example at min 2:10, why we didn't use the quotient formula?

[@MilitaryConnect]

+morearty Because you treat the numerator and demoniator seperately. You're not differentiating the whole expression. Let the numerator = f(x) and Denominator = g(x). Both functions can be differentiated simply, without product, chain or quotient rule. If the functions were complex, then the rules would've been applied.

[@articcircleado]

You don't take the derivative of the whole expression. It's a new rule that says "Take the derivative of the top and bottom separately until you get something in a determinate form."

[@ayeshaanuruddha]

L'Hôpital really teach me here to never give-up. and you too sir!

[@IdesireCake]

This one is in my Adams calculus book..

[@FatefKhan]

So now you can cheat from here. 😒

[@mathheadmcawesomesauce4325]

Sorry you're frustrated! He was merely taking the derivative of the numerator and the denominator using the chain rule. If you are feeling frustrated you should watch some videos on the chain rule and trig functions. After that, come back and watch this again. It should make perfect sense. Great job once again Sal!

[@GuruprakashAcademy]

good one

[@KhmerInception]

Great explanation, thank you!

[@chrisrock1990]

@piggygobyebye i don't know if i am right or not but i believe its because we are NOT taking the derivative of the whole function. we are only taking the derivatives of the numerator and denominator separately. Because that is what l'hopital's rule tells us to do. so what im trying to say is we are NOT finding f'(x) of the function f(x) = ( 2 sin x - sin 2x ) / (x - sin x) if we were we would use the quotient rule

[@MrCupcakeization]

how come at 3:21 you don't factor out a 2 from 2-2/1-1 and make it 2(1-1)/(1-1) and get 2 as your limit?

[@randomdd123]

I don't know if anyone else has mentioned this, but an alternative is to use L'Hopital's rule backwards and use the antiderivative. This would give you lim x->0 = (-2cosx + .5cos2x)/(.5x^2 + cosx) = (-2+5)/.5 = 6.

[@isavenewspapers8890]

I'm assuming that the first equals sign in your comment is a typo. Anyway, this is incorrect on several counts. Firstly, -2cosx + cos(2x) / 2 is not *the* antiderivative of 2sinx - sin(2x); rather, it is *an* antiderivative, an element of an infinite set of antiderivatives denoted by -2cosx + cos(2x) / 2 + C. Similarly, antidifferentiating x - sinx gives us x ^ 2 / 2 + cosx + C. In fact, to avoid confusion from here, let's label the first C as C_1 and the second C as C_2. Secondly, this is not how L'Hôpital's rule works. If you want to let f(x) = -2cosx + cos(2x) / 2 + C_1 and g(x) = x ^ 2 / 2 + cosx + C_2, then sure, that's fine. But if you just make the constants 0, then lim x -> 0 f(x) = -2cos(0) + cos(2(0)) / 2 = -3/2 and lim x -> 0 g(x) = 0 ^ 2 / 2 + cos(0) = 1. This fails the requirement that these limits should be both either 0 or ±∞ for L'Hôpital's rule to work. We can fix this by letting C_1 = 3/2 and C_2 = -1, but the question now becomes: what was the point of all of this? We just gave ourselves an indeterminate form that we cannot evaluate. Thirdly, you didn't evaluate your own expression correctly. lim x -> 0 (-2cosx + cos(2x) / 2) / (x ^ 2 / 2 + cosx) = (-2cos(0) + cos(2(0)) / 2) / (0 ^ 2 / 2 + cos(0)) = (-2 + 1 / 2) / (0 + 1) = -3/2.

[@kingkarl12341]

this is awesome i love the colors

[@emoflix]

Isn's it much simpler to cancel sin x and get -2 + 8cos x at the 3rd step? One step saved and no need to differentiate again, and the limit evaluates to 6...

[@alternate241]

yeah you could, and it might've been a good idea for him to mention that you can use trig identities to shorten the process when possible. But I suppose he really wanted to demonstrate how this rule can be used over and over until you get a valid answer. If he used a trig identity to finish, that may confuse some students into thinking they'll always need to use identities to get the right answer. That can be a daunting idea for students who haven't memorized them all yet. Baby steps!

[@tijnio777]

+Bitan Nath Yeah, it's what I would've done too.

[@michaelanthonyabenales]

@Bithan Nath yeah, for students

[@tireironman]

great example

[@wolfsegovia]

Great Video! Thanks man :D

[@jilaniniyascp785]

Really helpful do more

[@cooltop1]

Nice. Anyway can anyone tell me the use L' Hopital's rule in a physics situation?

[@chamod-s8n]

thank you❤

[@hadesbearer]

Hey man, thanks for all your help this exam season... I never knew about iterating l'hopitals rule like that! Useful

[@Smullet90]

It's actually a lot easier than you might think.

[@erinaerina8384]

I wrote Hospital's rule everywhere in my assignment 😆

[@Algo1]

would it be true that f(x) = -2cos(x) + 8cos(2x) / cos (x) i.e the function that solved the problem is the third derivative of f(x) = 2sin(x) - sin(2x) / x - sin(x) i.e the initial function?

[@isavenewspapers8890]

No. We didn't take the derivative of the overall function; we took the derivatives of the numerator and the denominator separately. If we wanted to take the derivative of the whole thing, we'd need to use the quotient rule: d/dx[f(x) / g(x)] = (f'(x)g(x) - f(x)g'(x)) / g(x)^2. But what if, by sheer coincidence, (-2cos(x) + 8cos(2x)) / cos(x) actually happens to be the third derivative of (2sin(x) - sin(2x)) / (x - sin(x)), even though we didn't calculate it properly? In other words, what if we have a howler: a mathematical line of reasoning that is invalid, but still gives the right result due to pure luck? As it turns out, no. This is not the case. I had a computer find the actual third derivative of (2sin(x) - sin(2x)) / (x - sin(x)), and it is not the same thing as (-2cos(x) + 8cos(2x)) / cos(x). I was going to actually show it to you, but it's a very long and nasty expression. Nevertheless, if you're still curious what it is, you can use an online derivative calculator to see for yourself. Now for some unrelated notes: When you wrote "-2cos(x) + 8cos(2x) / cos(x)", I assume you meant (-2cos(x) + 8cos(2x)) / cos(x). Division comes before addition in the order of operations, so you need to use parentheses. The same applies for the other function. Also, you used "f(x)" for both functions, even though they're not the same function. This is confusing; it's like if someone wrote "x + x = 4", but they had the first x be 1 and the second x be 3. I would recommend using "f(x)" for one of the functions and "g(x)" for the other.

[@itzbahri]

Never give up on L'Hopital's rule. We must keep fighting! I just watched 300 :L

[@BortVoldemort]

thanks so much!

[@ThePrmv]

You can use trig identities to get rid of terms that are bugging you

[@saadijaz463]

This is pretty fun to watch at 2x speed

[@salmanahmad1]

Million thanks

[@Darkknight9035]

but how much should i check for the limit? you differentiated the initial functions like 4 times before finding the limit.. so am i supposed to keep on going for 10-20 times to find a limit?????????????

[@hedonism13]

@mdwael haha, I know, I was just saying I thought that Sal's voice doing a gaming walkthrough was a funny concept. I wasn't making a request or anything.

[@theavantika31415]

@piggygobyebye I'm not exactly sure here, but I think it's because it's fprime(x)/gprime(x) soooo that's two separate functions. it's not the derivative of one function, it's the derivative of two separate functions (top and bottom). if that makes any sense... i'm not sure if that's right

[@amressamashour9223]

very good video

[@AL-go2mv]

Thank god for sal!

[@piggygobyebye]

Why didn't you use the quotient rule when evaluating the derivatives? I thought you had to when taking the derivative of a fraction...

[@DaelinZeppiTheComputerGamer]

Thanks for explains this so simply! For some reason my lecturer tried to explain it via the Taylor Series... This explanation (and in your previous vid) makes more sense to me.

[@psammyproductions]

gosh you're so good

[@yveeh13]

So does it mean that u have to do L'Hopital's rule 3x to get an existing limit? I mean what if for the 3rd time, the limit gave a 0/0 answer? would u finally say it doesn't exist??

[@tomphillips6743]

How do you know when to stop??

[@DJ1Leffty]

Wacom perhaps? Or other form of tablet

[@gairick9]

For all CBSE class 12 students... you cannot use that in class 12 board exams

[@OfficialSilverMoon]

Why

[@Heatcheck30]

@piggygobyebye I'm not sure if he explained in this video, but using L'hospital's rule, you evaluate the derivatives of the numerator and denominator separately. I know this is 3months late, but hopefully this explains this question to other viewers. MaTh RuUuLeZzZ!!

[@garrettlanzoni4809]

thank ya sir

[@Mirzly]

Thanks a lot :D

[@Peter_1986]

Is there any way to determine how many times you will need to use L'Hôpital's Rule for a given expression? For example, could you perhaps calculate at which derivative the denominator no longer becomes zero?

[@tarannumsk761]

When you don't get 0/0

[@isavenewspapers8890]

Yes, you can go through the process and find out. If you mean to ask whether there is some shortcut to find out without actually doing it, then I don't really know.

[@AtoHenok]

That's what I call 'Derivative Inception'

[@KISHAsodmg]

OMG thanks you

[@walete]

do you not have to use the quotient rule?

[@CaitiBridget]

Not in this case, because you're not finding the derivative of f(x)/g(x) - you are finding f'(x)/g'(x) (I'm just ignoring the limit for now to make it easier to type, but that's from l'Hopital's Rule). For example, if I have f(x)/g(x)= (x^2)/(x^2+x) The derivative of that is founding using the quotient rule. It'll come out to be d/dx((x^2)/(x^2+x)) = 1/(x+1)^2 Alternatively, if I said to find f'(x)/g'(x). You'd find the derivative of the top and divide by the derivative of the bottom. That should give you 2x/(2x+1). So to sum up, basically the rule says to find the derivative of one thing (f(x)) and divide by the derivative of another (g(x)) and so no, you do not need the quotient rule. I hope that's clear (=

[@thedramallama3089]

Basically you have to find the derivatives of the two functions separately in order to meet the criteria for L'Hopital's rule

[@janenikiita7919]

I love it

[@John-nd7il]

*take it to the limit by the Eagles distractingly is stuck in head for next hour*

[@mikaylason4370]

You are a god.

[@jeenyus720]

Why does cosX sometimes equal 1 and sometimes equal 0?

[@AtoHenok]

Dude, he has a graphing tablet. The ones with pen. I have seen his behind the scene video on TV

[@Iarabbro]

how many times do you keep doing it until you know that the limit dose not exist

[@hedonism13]

Am I the only one who think Sal would make great gaming walkthrough videos?

[@ramkrishna1404]

You have potential to become voice artist for anime

[@kmanalpha453]

Ik this video is 7 years old...but the [INAUDIBLE] word at 3:31 is "stage"

[@rakib17874]

What did u mean? I'm curious ..

[@fishboneisredhot]

after how many attempts do we give up on L`hopoital`s rule? I can`t imagine a problem in a test requiring a 10 fold L`Hopital`s rule solution

[@richardlauz1905]

Life Saver

[@the_growth_mindset.]

At 5.15 could you not of simplified 4sin2x to 8sinxcosx and cancelled out the sin x?

[@deathdeath1993]

man out-tiers math

[@sxntixgxs]

Parceeee, te amo.

[@DaSmorez]

It depends on the value of x.

[@ilovedancingxoxo1]

It with some kind of pen pad that is connected with the computer.

[@vaishakhsudhakaran1515]

Can anyone tell me the video in which he explains the case of 0 times infinity?

[@RMIDRIS]

nice vidio

[@milroxsox]

Why didn't my calculus teacher show us this on the first day?!

[@TuMadreCon]

L'Hopital's Rule in a nutshell: We must go deeper.

[@davidkim0678]

..so you have to keep using L'Hopitals rule until theres no more indeterminate? D:

[@Bin20259]

tanx mann!!

[@anthonyshea6048]

I don't understand. If you can keep taking the limit of the next derivative, do you just have to know to stop when you don't get an indeterminate form? Because if you took the derivative of the limit of last expression you would get 30 and 30 does not equal 6.

[@prashantchand2590]

So cooool

[@pisanghangus2]

mind blown

[@Ceecee384]

arent you supposed to apply the quotient rule? how can you just take the derivative of everything individually

[@priteshrathod9896]

Hey i have some problems in sums can you solve for me?

[@priyathh]

cant we just simplify 2(cosx - cos2x)/(1-cosx) to give 2(2cosx + 1) without repeatedly using L'hospital's rule

[@HeierMr]

mind=blown

[@firstson1]

and we're done!

[@khwajamahadhaq2743]

How would one figure out, When to stop? Like, Ummm...When, Enough is enough, So to speak of....