Chain Rule Examples

[kimberlyssy]

i love derivatives !

[TheRealCardshark87]

can this replace the formula method of finding the derivative of a function?

[ekar707]

You will go to heaven

[Tranquilance]

Anyone else thinking if they had khan academy when they were ten they would be in a better place in life right now? T_T Be grateful kids.

[jocktheglide]

Shouldnt you have simplified the 2nd line on the 2nd problem? Multiply 5 (15xsq2-7)????

[pink2bluee]

@jocktheglide You could, but it's the same thing. Depends on what your teacher/professor wants your answers like.

[jocktheglide]

@pink2bluee

yup gotch, some teachers want the simpliest down to everything simplified vs what sal has.

[UTUBES4]

Im only age 12 nd learnin calculus now, nd ur just makin my life easier nd easier :D

[HostingReviews1]

he's really good

[TheButterflytatoo]

you are awesome let me just say

[boberoi100]

you're great .

[MuhZubairAli]

I have no word to thank you becoz I want give you very very very very big thanks.May God Bless You....

[0verlordQc]

Thanks for all your videos! Very helpful in exam period!

[5daysnotice]

you sound like the guy of the old spice advert

[gideon1103]

dude..you are GOOD



[H0oLIgAnSsRo0lMaRrSz]

gosh ur much better than my teacherteacher

[lespaulbass]

chainception

[Alive4Metal]

Omg... I actually got that.... if my University professor had attempted to teach me this then there is no way in hell i would have followed so clearly.

[Clevername12]

oh my god i love you

[Nelsonpart]

this shows a major flaw in our education system.  Any schmuck can become a math teacher. However, why does every school need it's own designated math teacher when with the use of technology, everyone can learn from truly great minds such as Khan. Granted having a teacher to assign work and answer questions would be useful, but actual learning should come from stuff like this.

[scuzzulus]

@Nelsonpart learning comes from actually being motivated and working exercises.

[Nelsonpart]

@scuzzulus So, you are saying the quality of our teachers is irrelevant because learning is an independent process? Wtf is wrong with you.

[scuzzulus]

@Nelsonpart no fool i`m saying that it doesn`t matter how good the teacher is if the student doesn`t want to learn.

and you can`t learn math by watching videos. you have to actually practice. its like trying to learn tennis by watching wimbledon. doesn`t work.

of course the quality of the teacher is important, but its not as large a factor as many think. motivation + any decent textbook will get you there.

[Nelsonpart]

@scuzzulus sorry to offend ya Mr. Hunting......

[scuzzulus]

@Nelsonpart you'd better be. i was about to reach through this screen and smack your face against your keyboard. but i shall relent due to the fact that you repented.

[Nelsonpart]

@scuzzulus dude, you're an idiot....you have no idea what my previous comment meant...thus, I rest my case. don't reply to this. btw.. If you were to reach through my screen with violent intent, I would swiftly chop off your hand with the machete hidden in my pants. no more five knuckle shuffle for you.

[scuzzulus]

@Nelsonpart bitch i know exactly what your comment meant - its a reference to the movie where mat damon plays a savant who teaches himself high level math. anyways i have no beef with the khan academy or anything else like it, its a great resource for all those who need it - but it won't fix education and it won't make people WANT to learn. these are facts.

[pon00050]

@Nelsonpart

I presonally think that the role of teachers in school

should change to that of coaches, more of a person

to help students be self-disciplined.

[theonetrueac]

@pon00050 Agreed. Just like a coach they can tell you how to do it, but they force you to want to do it enough that you actually commit to doing it. Kids need a reason to want to learn this stuff though. I remember being a home-schooled high school student who didn't do hardly any schoolwork. It later came to haunt me when I decided that I needed a degree to get a good job. Now I struggle with this Math @ 25, when I should have learned it years ago. Teachers should motivate as well.

[lentzdorcely]

Khan you can fuck my bitch

[SunshineK321]

You are my hero

[monday2471]

thank you so much for doing this!

i can never learn properly in school because my mind wanders a lot, and if i miss even 1 thing the teacher says, i won't understand the entire lesson!

but i can always replay videos and that's the best part

[maquillage11]

WHY DO I GO TO MY CALCULUS CLASS? I LEARN NOTHING. thank you soooo much!!! i'm passing calc because of you.

[IamAware26]

you should be the next president. thanks

[Browneyyes]

Very nice! It was not confusing at all, thanks for doing a complicated one!

[MrSamarthz]

you didn't confuse us ...you just cleared all our confusions !

[MrKushalrox]

dude u justify the quote that says " I bunk school to study in peace". thanks man . great help

[Sleepyabess]

i have a question-at 4:35 the fourth n last step of the equation-is that the finished result? would be happy about getting an answer- thx to khan academy by the way

sry for my english

[monkeygirl82000]

Not gonna lie, when you said that that example at 8:30 was going to be one of the harder ones we'd see on our test, I freaked out--you obviously haven't met my teacher. I hope you'll have examples with trigonometric and/or logarithmic expressions!

[saminm1]

I'm 16 and i understood this rule.

Thank you so much.

I watch your videos because they are entertaining.

[ChrisCummins]

A chain within a chain... Inception, eat your heart out!

[DRCRAZY03]

@ChrisCummins chainception :O lol

[glitchdood]

nvm lol

[glitchdood]

@7:40 is that last orange line multiplied by the entire line above it? or only -6x^-3

[publicanimal]

I like to start from the outside and work inward but you get the same result either way.

[liyajohnsphilip]

i been struggling in calc this whole time n i tried my best to learn my text book n the more i tried the more it confused me but after watching your videos for d first time itself it all made sense...THANK YOU SO MUCH.. I DONT KNOW WHAT I WOULD HAVE DONE IF I DIDNT FIND YOUR VIDEOS. :)

[MaMahmod]

I am Arab and you are really really really good at teaching ,thanks alot for this

[Ruth2475]

thank you! thank you! thank you! thank you! THANKS YOU!!!!!

[Student1304]

Your a genius, you have helped me sooooo much. You make maths enjoyable.

[buzzy918]

I was confused for a second but once i rearrange it in my head to show the -6x^-3 on the left instead, it made total sense

[Chat2Conaz]

WOOHOO! Thumbs Up! if your the 100,000 viewer! I love Khan Academy and these videos!

[LemonTree224]

DIdn't confuse at all! Thank you!

[topik627]

jazakum'ALLAH'UKHAYRIN! may, GOD bless you and I eveyone else watching these videos!

[geniusAyushPateria]

Super easy

[erin210992]

im studying egineering and i have no idea what i'd do without these vids...

[MrKelvint56]

ooooh i got you

[MrKelvint56]

why did you get to power - 8 instead of to power - 9

[heynowwilliamson69]

Making math fun and easy again for me, thanks!

[TarroYoko]

The example at 5:56 reminds me of INCEPTION, but with derivatives, anyway thanks so much as usual!

[goofguy316]

YOU ARE A MESSIAH..... Of my marks, thank you so much!

[datxcod]

I don't understand why he doens't know how to erase the screen in the previous lesson but in this lesson he does it easily, it's like he didn't bother to check if the program was able to erase the screen before recording the video... 0_o

[Huttate1]

@datxcod That is what you take away from this?

[gitrdone300ex]

I have spent days trying to learn this and you taught it to me in 10 min. Awesome videos. Thanks for the help.

[pwnions08]

LOL i was thinking "you bastard!! " when u put that ugly one on the board.

3x^-2+(5x^3-7...)^3 ) .. Thanks for all your help thus far

[boyette2001]

man, you are saving me repeatedly in calc 1..really appreciate all the work

[kream013]

i agree with all these people who find this video soooooooooo helpful....i was trying to study this in my calc1A text book and the longer i tried to understand the chain rule the more confused i got. this video makes it so clear. thanks a lot.

[assadullahch]

AOA ..... Thank you so much sal... u r great

[rotflmaopmpqxyz]

so to find derivitave of "blob" raised to a power you just pull one "blob" out and find the derivitave of "blob" and then multiply it by the derivitave of what you are left with when you pulled one "blob" out.

Is that right?

[ellayxoxo]

"oh look at that, that was pretty fast"

aha, i love how your lessons contain some personality & scomments as opposed to strictly boring lessons delivered in a monotonous ways (a trait most teachers seem to posess). I genuinely enjoy your videos as they are didatic and interesting!

[numberJ5]

does the first chain go on forever? How do you know when to stop? lets say you stop at the answer where the 2nd term is raised to the -8. so, you could keep subtracting one indefinitely? 

[LAnonHubbard]

I recognised Paint (pbrush.exe) when you dropped down the Image menu. It's amazing you were able to do so much with this back in the day. I see in your more recent videos things have got a bit more sophisticated and tidier. Nicer colours, curves, and drawings. These videos are still fantastic though. I assume you've always used a pen and tablet (I know you do now as I saw the Bill Gates video). I just tried writing using a mouse and it's pretty impossible.

[HeyitsJohnNerdBlog]

You have inspired me to take Pre-calc over the summer.  So during my Junior year i can take AP Calculus. thank you!

[pro4skill]

amazing... I watched these videos non-stop, and now I'm going to be ahead in calculus! (I'm in precalc)

Thanks so much!

[priyakoko]

I LOVE YOU!

[majingwei]

Khanacademy saved my life, period. 

[greenfish13]

wow i haven't looked calculus for almost a year, only business calc at that, but this is coming back fast!

[lotslepy]

the world needs more people like you, sal.

[MrZia324]

great work khan. 

[zeozionistic]

wha.....

[TylerAAADurden]

that's not confusing at all, but simplifying that expression is gonna be a bitch :D

[mzdbnell]

This helps me more than my professor. Thankyou sooooo much.

[michele7656]

2:25 "To the minuuus 8." I don't know why, but something about the way you said that was hilarious. Thank you for making these videos and being awesome.

[Patambo2000]

SAL..... "your starting to go nut's"! hahhahahah

YOU ARE A GREAT MAN SALMAN!

Eres una gran persona hermano Sal, gracias por existir!

[omgim18]

i love you. u are so much better than my math prof who just doesnt even make sense. i think u are going to be the only reason y i pass my midterm tomro. THANKS. u are doing an amazing job. keep it up!

[jgrimes5353]

Not confusing at all, thanks for making this easy to understand

[yuristatus]

Dont apologize-- u sooo didnt confuse me ! ^-^

[sadiq5519]

You are so amazing, thank you and keep up the good work. I will try to send you something helpful to your personal message in exchange for this favor. thnx

[Abz90L]

wooooooow, THANK YOU!!!

[david0aloha]

I like the videos, but I find it much easier to think about it in terms of a function of a function (of a function). eg. f(g(x)) or f(g(h(x)))

For a simple one:

dy/dx = d/dx f(g(x)) = f'(g(x)) * g'(x)

For a more complex one:

dy/dx = d/dx f(g(h(x))) = f'(g(h(x))) * g'(h(x)) * h'(x)

For even more complex (like that example):

dy/dx = d/dx f(x + g(h(x))) = f'(x + g(h(x))) * (dx/dx + g'(h(x)) * h'(x))

Maybe not the best notation to learn first, but makes solving ones like that last problem easier.

[soloohk]

its a really complicated example ! i am wondering if we can find this polynomial equation in real life situation.

[cthep]

This guy and PatrickJMT should battle.

[whisper012]

@cthep HAHAHAHA YES!!!

[omarhea1]

How do you solve this? 4x^2(3x-2)^5

[whisper012]

@omarhea1 If by solve you mean 'find the derivative' then it is: 4x^2 (3x-2)^5, first you find the derivative of the inside, which is 3. then you multiply this by the derivative of the whole bracket which is 5(3x-2)^4 then the normal derivative rules apply. so you multipy out 4x^2.3.5(3x-2)^4 which simplifies to 60x^2(3x-2)^4. Can anyone confirm this is correct??

[boropyt08]

WaiT!! I hAVe a QuestioN....in the last step of the first problem, (around 3:15-4mins) why did you multiply the -7 by first half of the problem instead of the (x^3+2x^2-x^-2)<<--second half of the problem??? or does it matter?? Help ASAP PLZZ!!

[laram24]

@boropyt08 Because you would have to decompose the whole thing that has an 8 as an exponent (thus multiplying the last expression itself 8 times). So, it is way easier to distribute the (-7) in the first simple polynomial expression.

[dhn09]

@boropyt08 because it doesnt matter what side you multiply it too. Its multiplication, if he were to do it to the other side the answer would look different but if he were to simplify it itd be the same

[whisper012]

@boropyt08 Because '-7' was a multiple of the whole thing. It may have been easier if he expressed it as -7(3x^2+4x+2x^-3)(x2+2x^2-x^-2­)^-8 . the reason you can't multiply out the -7 with the second bracket is because it is raised to a power. to do this you first would have to multiply out the whole bracket by its power, which when it is the power -8 .... Yikes :S ... which is the reason we use the chain rule in the first place, so that you don't have to multiply out the bracket by its power :)

[Felicity1616]

thank you for doing these!!!

[lbrtdy]

I got a 102% in pre-cal with your help, now its summer time and learning calculus too! Wish me luck on AP Calculus the coming year hehe :)

[leadershipcouncil]

You should be a math teacher at my school. Seriously!!

[TheRichestCadaver]

Excellent explanation and I learned a lot, so much that I could catch a mistake, somebody also did before me.

Final answer: [(15x^2-7)(5)(5x^3-7x)^4-6x^-3­] (3)[3x^-2+(5x^3-7x)^5]^2

[predan2k]

wow! 5

[Aukatrau]

I'd give you 5 stars but I can only give you a thumb up. Thanks for explaining stuff in such a clear manner. You are going to be the sole reason I pass AP Calculus.

[bloodybigballz]

i actually followed through the whole problem, thank you.

[richy70blitz]

Khan The Greatest, The legend, The Greatest Cyber Teacher Thank You For Your Time

[442Alec]

Sal, you're starting to go nuts, this is gonna take us forever

[michelvellve]

Nice tutorial!

It really helped me with my homework :D

Thanks for sharing your knowledge xD

[nixenberq]

Yeah man, you are awsome, thank you for your lessons !!! Thought I can't learn these things :-D, but its quite clear now, you saved me a lot of time...

[apricotsFTW201]

I love you. lmao

[browneyedgrim]

my current calculus teacher is a Dr. and the head of my school's math department. my school is a nyc specialized high school, with a strong focus in math. you've taught me so much more in like half an hour then he has in the past 2 months.

[allioopla]

Nope, no confusions, You are simply amaazing. I Passed pre calc because of you and now i'm on calculus. Thank you for being such a great teacher. and Thank you for posting these for free!

[mooseway86]

You just saved me $2000 by not having to attend summer classes!

[ahahahcrap]

@mooseway86 you should donate some of the money you saved to the khan academy!

[mooseway86]

If only i wasn't out of state and broke as it is :(

[milesbennet]

@mooseway86

62thumbs x $2000 is? :D

Thanks Sal!!

[milesbennet]

@mooseway86

Plus your thumb! 

[WarmBread123]

if you do what you said, do you still have to do the chain rule again? it's only decreasing the exponent by one.

[dawitb111]

@WarmBread123 u're stupid

[crossfuse]

this is soooo much easier for me to understand...i havn't been getting good grades in calc and i think im finally getting caught up

[siriuslyD]

you're so much better than my current calculus professor, THANK YOU!

[johncharlesvii]

@siriuslyD im there with you

[pakoluevano]

dude you are the best math teacher ever man.... good job i get this stuff man thanks.!!!

[picarolus]

I have been intimidated by the chain rule for a good year now, and I finally feel like I'm getting it. The different colours help! Awesome job. Brilliant thing to do with a youtube account :)

[cocolombo]

Great videos. Thanks.

[sabrinakbh]

A good teacher makes a world of diference.

Thank you!

[johnyori]

Why does the seven in the first problem 3rd step multliply against the first parentheses when it is completly on the other side of the mult sign???

[zardinuk]

cause you're just multiplying three things together, the first parenthesis, the -7, and then the second parenthesis, you just have to make sure you don't screw up the order of operations. When it comes to multiplication though, there is the commutative property, which says you can multiply things in whatever order you want... just take respect to that -8 exponent on the right, if you were to multiply it with the second parenthesis, you'd need to expand the -8 first cause of the order of ops. Ya?

[mrbinja]

Thank you for saving my butt! My teacher doesn't speak much English and when she does, she just reads from the book.

[datCRUNKbuay]

Joking with the below comment obviously, but seriously i have been going to private schools (very expensive for my parents) since i was a little kid, and have not had a teacher that was as patient and easy to understand as you!

[datCRUNKbuay]

Will you marry me? I am a 20 year old straight male.

[MsKb1992]

god bless you!

you make one who hates math turn into a math lover:)

congrats for finding a faster way in erasing:)

take care,WE need you.

[olzorz]

if every math teacher were like you, i guess the world would have been a better place!

Greetings from norway

[prs2106]

I presume those match teacher who thought us will also desire Khan as teacher ;) ?

[aegisforex]

Interesting. Every calculus book I've ever read says to start with the outside and work in, but you do it the opposite way.

[benoc1402]

one question though... how can one human being know so much stuff. his head must be ready to pop!!

[Liist]

Lots of human beings know so much stuff. Only difference is that Sal Khan is one of the few that also have a soul, unlike your professors.

[nriab23]

@Liist no truer words said,

[benoc1402]

the guy is a genius..

[algeiranall]

this is such a gr8 expanation- i get it a lot more...thanx for the tips!

[edmun09]

dude your vids are freaking awesome u are THE BEST MATH TEACHER EVER

[flash452]

dang i'm learning this on my own since i can't take calculus at school i get it, thanks man

[jtyson91]

dude you're a future saver. thank you!!!!!!

[nastynix2]

thank u!!!thank u!!!thank u!!!thank u!!!

im *jumpin with joy*

[XsmashX1]

Dear Sal,

You're a life-saver.

Sincerely,

Jimmi

p.s.

You're 100 times more lucid and cogent than my professor. You didn't confuse me at all.

[moniquebabby]

it gettingharder and harder but thanx 4 da explain

[tinalafermy]

In the last example what happened to the -6x to the negative third?

[policeman1313]

what do you mean? It's the derivative of 3x^2, and gets added with all the other derivatives in the brackets on the 2nd line.

[supercoupon2]

was waiting on that last set of brackets :) Great job as always.

[rinwhr]

This guy went to Harvard.....

[policeman1313]

huh?

[rinwhr]

the person who made this video.

[jgonzalez229]

awesome teacher ! im currently taking a precal class but its too slow so im learning as much as i can and you help so much

[MAURICIOBRAZZILL]

you did confuse me with this last example, but you're doing a great job, I understood, thanks

[banditdk]

UR a super great teacher man thump up....

[Rubia715]

Have I ever told you how much I love you?

[johnmchen]

who cares about his ethnicity.. mathematics is a universal language

[rockapejr]

My professor in college in Indian and sounds nothing like him

[jerannamo]

regardless of race, your doing a great job and really helping me out. i have quite the incompetant teacher who i just realized has been teaching us the derivative equation incorrectly... but i'm glad you are here to help. please keep it up.

[Nutterbutterz95]

I lol'ed really hard when I read that he was teaching you incorrectly. What equation did he use instead?

I'm genuinely interested.

[doggystyledave]

I don't know why it matters, but everyone says this guy is from India... It sounds like he is a white guy with his nose stuffed up.

"all of that" over x + 7.... lol.....

[alankarmisra]

It's amazing how the simple use of colors makes this so much easier to visualize (of course it takes an awesome teacher using those colors to drive home the point :-) ). Thank you so so much! This whole series of lectures is A-W-E-S-O-M-E (I wish I could color them all different ;-))

[mrmicco45]

all of these videos have been extremely helpful! i took calculus this past year and actually did well, i just wanted to refresh myself on the basic stuff before i head off for college this fall...you definitely are an incredible teacher! thanks a lot