Chain rule | Derivative rules | AP Calculus AB | Khan Academy

[@Fish-vs6jf]

This left me far more confused than when I started

[@montvmontv117]

my prof said with chain rule to always say "the derivative of the outside times the derivative of the inside" it may make sense. so (stuff)^2 derivative is 2*(stuff) then take derivative of stuff. so 2(stuff) * stuff ' here sin would be cos. 2sin * cos

[@priyankaboopathi729]

Exactly.. 😥

[@Saurabh_Tewari007]

Well it cleared my doubt even though he wasn't really teaching

[@aimzf20]

MonTV MonTV this cleared it up so much for me, thank you!!!

[@sudhamajayanthi]

@@montvmontv117 I understood nothing but “stuff” 😶

[@Yue27s]

wow super thanks mr.Khan kumar, now i really understand with this thing called chain rule how it work, but still cant visualized it but thanks i clearly know how it works now

[@jimshtepa5423]

how does one tell composition of functions from single function?

[@beegking]

this is so clear

[@rosskious7084]

Master explanation !

[@Eeeurrrrrr]

I praise thy holy salaman kotal khan for his great work in the field of mathamatics. Without his holy butttery voice, I would be but a mere padwan of what I am today

[@naziasiddiqui8807]

U play mortal kombat

[@Kervvy]

@@naziasiddiqui8807 U play DN

[@EastSideGameGuy]

I sometimes come here to listen to his voice ._. it's soothing makes math Beautiful

[@bonafidecatlover3443]

Coming here for the first time and I can already see this man having a wave of fangirls

[@Jcava5]

sorry, i feel like you shouldn't have used sin x, and instead just a simpler function. ive learned and understood the chain rule before and came here for a refresher and this didnt help :(

[@jamessanpedro223]

(x+y)^2

[@PenandPaperScience]

Or ln(x^2), like I solve step-by-step in my newest video. It might help you understand :)

[@chawza8402]

oh my god! my teacher taught me this but never mentioned it's called chain rule. I blindly use this method without knowing what it is for!

[@nafy2302]

What is wrong with your teacher

[@chrisfreilich]

Still trying to figure out EXACTLY what 'with respect to' means. People always seem to gloss over it or talk around it.

[@AFCA-vn9bl]

If you differentiate x^2 + y^2 with respect to x you will get 2x while differentiating with respect to y gives 2y

[@thanaa6401]

It's just that you want to find the derivative (i.e. slope of the curve y = f(x)) AT the point x So whatever the point x on the line x squared, it's derivative will be 2 times this point ( i. e. 2x) When I want the derivative of x squared with respect to sinx. I mean this: whatever that point x is, I'm gonna find its sine. the specific value of this sine, I will input it into x squared and see what will be tge slope of line on THAT value. So, I take x, I input it into sin function, I get an output, I input this new output into the x squared function, I get a brand new output, and I want the slope of the line on that specific brand new output. I encourage you to watch composite functions videos from precalculus playlist + revise the intro videos on derivative as a concept

[@satwikshourya2199]

It means that how does one value change with respect to another meaning , how much y will change if x is changed, for example if you derive postion x with respect to time t, then you get velocity meaning that derivative of x with respect to t is v, so when t has changed a little how much did x change or rather by factor did x change, that factor in this case is velocity v

[@fantastic_jarod2391]

Yeah lowkey that explanation kinda sucked first L khan academy vid ive seen

[@p.c2750]

What is the problem with some college/university teachers. is it laziness? I guess it is pride - it seems they dont want subjects to appear simple, too simple for students to suddenly catch what they spent many years to train for. It is not only right, but an honest thing to "call a spade a spade." Thanks KA!

[@Jorge-125]

Everyone in the comments talking about math stuff while I'm over here trying to count how many times he said respectively. I'm just over here thinking how much respect is in this video.

[@lidestudios5094]

I only watched this to review, I already know how the chain rule works, but the way he explained it makes it look more complicated than “derivative of outside when x= inside function, times derivative of the inside”

[@MatthewReader]

Any good videos that you thought were a better review?

[@thanaa6401]

He just doesn't want us to memorize it thats why he took harder approach

[@josuequintero-td5gh]

Yeah because he's trying to make you understand why it works. Otherwise you're just memorizing formulas without knowing what they do.

[@oferzilberman5049]

I didn't quite understand what does it mean "With respect to something", And quora and google didn't really help, Can someone help me on this? Edit: I figured it out and it turns out it just means it's the x axis but now it can be substitued as other variables like t, So when it's "with respect to something" it means that the y equation is using that variable as the x

[@alyssamanwarren8321]

i don’t either

[@Jorge-125]

He's paying his respects to the math gods.... Jk, people always use that term and I still don't know why.

[@PenandPaperScience]

I just uploaded a video where I go through such exercises step-by-step, explaining everything along the way. It might help you. Regarding your question, take this example. 1. The derivative with respect to x of the function x^2 is 2x. 2. The derivative with respect to x of (x+3)^2 needs the chain rule to be solve. 3. The derivative with respect to (x+3) of (x+3)^2 is 2(x+3)

[@enternamehere6686]

Would you mind not taking too long to get to the point please

[@littlejoestar4]

definitely more confused now 😭

[@abhilashmishra5397]

this video just saved me from failing in Physics. Thanks a lot Sal sir

[@rajingajadhar4135]

Found this channel in grade 12, and I'm coming back to it now in my final year (hopefully) of uni.

[@RemotHuman]

2:25 isn't it the derivative of the whole function with respect to the inner function sin(x) or are you saying that d/d(sin x) (x^2) = 2 sin x I think it should be d/d(sin x) (sin x)^2 = 2 sin x

[@ernstboyd8745]

in the case of a linear function (where the function is just multiplying x by a constant) the derivative is just the product of the two derivatives. if f(x) = 5x and g(x) = 10x. then. g(f(x)) = 50x. and (50x)' =. f'(x) * g'(x) = 50 if the outer function is not linear like g(x) = x^2 then the slope changes depending on the input it gets from the inner function f this is why the slope of the g(x) should be evaluated at the same value as it gets from f(x) in other words. (g(f(x)))' = f'(x) *. g'(f(x))

[@issac1071]

Hello, thank you so much for the video, but I have a question please: did the last notation was really treated as a fraction? Regards.

[@aimzf20]

No, it’s not a fraction. However, the way he write the notation can help you to think about the derivatives. They’re not fractions, however it may help you to think about them ‘cancelling’ in this way

[@issac1071]

Thank you Andy F

[@ernstboyd8745]

I think the chain rule is easy to explain as a 2 stage amplifier. so (g{f(x))' = f' * g'

[@anushreesolanki4782]

If x =at^2 and y=2at, then find d^2y/dx^2. help me with this concept

[@OppenheimerUpandcoming]

OMG I FINALLY UNDERSTOOD THIS THING THANK U SO MUCH

[@hloniphalisa4632]

I love you so much ,So wish you could be my math lecturer. you are better tan him. thank you so much I was thinking o changing my course but now never all your video you make math easy and simply

[@alep.2672]

I think this one needs a re-do by the same person

[@kenonkenkay9426]

thank you. your video is very clearly to understand, I'm sure it will help me pass my mid-term exam tomorrow. lmao

[@ValoriYT]

How'd it go?

[@spoonyabyss8814]

Ya dude how’d it go

[@ameen5586]

Bruh

[@Jorge-125]

2 years later. So how was your graduation?

[@Ndiedddd]

@@Jorge-125 3 yrs* 😂

[@anooby]

dang that actually made sense thx sal

[@BoZhaoengineering]

This planet needs you! Thank you for the learning videos!

[@qualquan]

The U substitution of inner x terms is simpler 2nd step is finding dU by differentiating U = dU/dx to express everything in x terms So 2 successive differentiations. "Outer and "inner"" Final result expressed as a differential coeff (DC) as a rule

[@tylergust8881]

It both makes sense and it doesn't make sense... But to be fair I'm still in trig... I don't know too much about the derivatives...

[@Syntaxxed]

all those hours in school vs a 5 min yt video... it keeps happening but I keep being amazed

[@sameerpokhrel4379]

When we take the derivative of the function f(x), we write the derivative as dy/dx but Sal wrote the derivative of h(x) as dh/dx . So ain't we supposed to write the derivative of f(x) as df/dx? Please clear my doubt.

[@Andrew-ok9gt]

It doesn't matter because f(x) and y are the same thing.

[@jamessamuel1255]

Can write it as either

[@jamessamuel1255]

h(x) can have the derivative h’(x) or dh/dx

[@No_BS_policy]

y=f(x) ->>> y is a function of x dy/dx ->>> derivative of y with respect to x, same as derivative of f(x) with respect to x or df/dx

[@GEORGINA-topic]

What did u say?

[@mohommedameenahmed2099]

Nice

[@kennyelliott9164]

Wouldn’t cos x actually be -cos x because the derivative of sin is -cosx

[@ranaAmeer9349]

Shouldn’t we just use the power rule to get 2 sin(x) and that’s it?!!!

[@jamessamuel1255]

Rola Ahmed yeah I’m confused where the cosx comes in and why

[@sophialaird6388]

Cos x is the derivative of sin x. It’s weird, but yeah

[@kshjnbvsmd]

@@jamessamuel1255 have you got any idea now, im stuck in the same confusion..

[@what5493]

@@kshjnbvsmd d(f(h))/dx = 2nd line of 4:10 it’s kind of law. If you differentiate it. You want to differentiate of d(f(h)). Firstly if f(h)= (sinx)^2. That means it’s differentiation will be(d/dx (x^n) *d/dx (x). put here x=sinx.

[@mattboatman5769]

Yeah, this is confusing me as well. I get that the derivative of sin x is cos x, but where did the second sin x come from in the first place?

[@yaredt4345]

What is happening here, the math is flowing into me 😁😁😁 this is witchcraft man 👏👏👏👏

[@davidsweeney111]

You do tend to present calculus in a mechanical way and so all the meaning and understanding is missing...

[@GhoshA]

David , hmm

[@ethanp4050]

stfu

[@snickerdoooodle]

"Mechanical way"? What?

[@stinkymans5002]

I actually agree with this. I get very lost in the “derivative x” “du dx” “x x x” He doesn’t give any of it meaning

[@hawkmoths]

well that's also likely because this is an intro to the chain rule. he is using terms that he likely already explained in a different video.

[@SUMANKARANTH-u8y]

The power/chain rule is the saviour of math

[@jonathanlionel3740]

first comments

[@kaylam.6264]

So I'm still confused as to what the answer is 🤔

[@freakinbeast]

I was stuck in the same question and he that that question only.

[@dazzdazzle8758]

I wish he had used a simpler function instead of sin x

[@becca5501]

dang, he did such a great job.

[@abrueck2518]

First comment

[@rafsanaakterjannat8066]

great work I don't know that about this

[@faythe03]

thank you!

[@xobk]

I recommend a new intro to this intro.

[@madisonroberts258]

Im gonna do great on my test tomorrow

[@mysteryspy7]

why he multiplied with cos x with 2sin x ................ i am stuck up can anyone please help me!

[@PenandPaperScience]

I just uploaded a video where I solve such exercises step-by-step, explaining everything I do along the way. It might help shed some light :)

[@mysteryspy7]

@@PenandPaperScience 👍

[@egregiouslytalented2381]

Love this

[@6-b-46bsricharan2]

That's amazing.

[@anamikasarkar8862]

You should have used different colour pen

[@ethencubberly5073]

respect

[@JJPMaster]

use the chen lu

[@l.m4631]

thank you so much

[@amitai8204]

very understandable! thank you!

[@konvictz0007]

Hi Sal, love your videos. Just need to update it to 1080p, will look sharper on bigger screens.

[@braden6973]

Watching this on a 27-inch 4k monitor, Never once noticed the quality. The only thing that matters is the content of the video.

[@oddwork2183]

Great

[@adityamukherjee9462]

I feel like I have already seen this video before tho.

[@sameerpokhrel4379]

Dejavu!!

[@bhubneshwarprasad9554]

Aumsum time

[@wasimakram-rs5db]

differential geometry a first course by d somasundran PDF mein knsi website sy mily GI???

[@bayesian2007]

Perfect explanation.

[@GhoshA]

Voice and presentation is good .

[@ahmetkarakartal9563]

2:16 did you make a mistake? the answer should be sin2x, isn't it?

[@chawza8402]

I think he is correct. if you see the upper example. you can replace the Sin x with the a, which is the same x but different x value. the first x is a, and the second x is Sin X.

[@jeremychamberlain6332]

You are both correct. By the double angle identity, sin(2x) is the same as his answer of 2(sinx)(cosx)

[@mattboatman5769]

Where does the second "derivative of sin x" come from at 3:06? I thought we had already established that the derivative of (sinx)^2 is 2sinx.

[@flying_asparagus1279]

It's multiplying the outer function times the inner function. If we see sin x as u, then the original function is u^2. So Sal took the derivative of u^2 and got 2u, which is 2sin x. Then he took the derivative of the inner function, sin x, and got cos x. Therefore, the derivative is 2 sin x cos x.

[@No_BS_policy]

Nah mate, 2sinx is just dh/dsinx. We are interested to know dh/dx which is equal to dh/dsinx * dsinx/dx. Hence, the chain rule is applied.

[@doooofus]

@@flying_asparagus1279 what i find confusing is why we multiply the differential of the outer function with respect to the inner function, by the differential of the inner function with respect to x. i must have learned chain rule years ago now, but coming back to it i realise ive only memorised how to apply it sort of mechanically and dont have an intuitive understanding in my head of why it works that way. Sal doesnt really explain that in the video but just gives us the easy way out of thinking about it in terms of the leibniz notation with the fractions, but even points out that this isnt mathematically rigorous.

[@dhruvsingh9243]

2:24 the differentiation of sin X will be cos X

[@janardhansharma9451]

It will be 2sin x as we are finding derivative w.r.t sin x

[@brugamer6031]

Btw this video doesn’t not help me because its very different with my school work as i know chain rule is UV-UV % V square

[@zalida100]

That may be a quotient rule you are confusing it with - Sal was explaining the chain rule.

[@bulasinha2772]

That's the quotient rule...