Using the Chain Rule - Harder Example #1

[adityaindia93]

I used to be terrible at maths like you, then I took a patrickJMT to the brain.

[NicoleAnnHosking]

Your voice got deeper over the two videos XD

[czarinadify]

how do you identify which one is the outside and inside when doing ln and e?

[veeralp95]

GREATEST MAN EVER! THX SO MUCH!

[chocobo1028]

Can you explain why you subtracted the exponent from the first time you did chain rule but the other times you left the exponent alone?

[Libservative79]

What happened to the sin(x^2+8x)^4? The 4th power just disappeared.

[StarlightOnly]

this is so easy :D Haha :) This is what I love about maths! 

[Philliesphan007]

Thanks so much man. You don't understand I have a TERRIBLE teacher for my AP Calculus class... in the beginning of the year I got a 57 on my first test.. after that a friend who graduated recommended your videos and since then I got an 84 on the next test, an 100 on the next quiz, a 94 on a quiz yesterday.. and I'm confident for once going into another test on Thursday. You are the best. :')

[patrickJMT]

@Philliesphan007 glad to hear that you have had such a nice turn around ; )

[darthmatisgod]

y=[tan(sin([x^2+8x]^1/2))]^5 y=u^5 u=tan(v) v=sin(w) w=a^1/2 a=x^2+8x dy/du=5u^4 du/dv=sec(v)^2 dv/dw=cos(w) dw/da=1/2a^-1/2 da/dx=2x+8 dy/dx=5(tan(sin([x^2+8x]^1/2))­^4*sec(sin([x^2+8x]^1/2))^2*co­s([x^2+8x]^1/2)*1/2(x^2+8x)^-1­/2*[2x+8] Thanks for helping me understand how to do this myself! (substitution is my favorite method)

[Monerasauce]

This is incredibly helpful! I'm studying for my midterm for tomorrow and you video finally got me understanding this rule. You're the best!

[patrickJMT]

@Monerasauce good luck on the midterm!

[metalhead6242]

the day you stop making videos is the day I fail Calculus

[patrickJMT]

@metalhead6242 ha, only if i then delete them all! : )

[hhira13]

@patrickJMT please don't! :(

[Bud7205]

Second step, you missed a close parentheses

[Moemenk]

@Bud7205 lol, oh no, the world is going to end..

[icecoldgangstaa]

Oh my god! It all makes sense!! Thank you! :D

[dragonflar8]

Why don't you put a multiplication sign between

cos(x^2 + 8x)^(-1/2) and (1/2)(x^2 +8x)^(-1/2)?

[muneco0211]

My Calc. professor makes us break it up into y=f(u) and u=g(x) and then take the derivatives separately. I get so confused and have to keep glancing back at the derivatives I took. Your method makes it so much simpler by keeping everything together! Is there an advantage to splitting the original function like that, as opposed to your method?

[patrickJMT]

@muneco0211 i think they are just trying to break it down and have you see things as a composition of functions. i also do not care for that method.

[weststep]

Thank you thank you thank you :) You, my friend, are my hero. I need calc for my bio major and pre-med ....and I am NOT a math person (I'm left-handed like you, so I was going to use that as an excuse but apparently it's not!) =)

[patrickJMT]

@weststep ha, happy to help. my phd mathematician wife is a lefty too, so no, that excuse will no work : )

[AceAites]

In "sec^2(sin(x^2+8x)) ]" <= I'm not sure if I'm just trippin' but I think there's an extra bracket at the end. Otherwise, your example was CRYSTAL clear and I completely understand how the chain rule works now. Thank you so much ! :D

[tennis1337]

Clear explanation step by step. Thanks!

[patrickJMT]

@tennis1337 no problem : )

[4znDude]

KhanAcademy and you are awesome except your handwriting is neater.

[kingallen08]

you are the MAN you are a genious

[patrickJMT]

@kingallen08 no, i am very hard working. a BIG difference

[IhateMissouriALot]

I see we got a nice real world example here.

[Morphae]

Do you have any tips for having neat handwriting, frankly i'm jealous of you.

[patrickJMT]

@Morphae write slower : )

[TheNumber2Pencil546]

my teacher subsitutes everything with a bunch of U's and im so lost

[patrickJMT]

@TheNumber2Pencil546 yes, i am not a fan of that method either

[jamesbond847]

@patrickJMT i am not understanding why you are taking the derivative of the function sin onwards according to your previous video shouldnt you leave it after you have the derivative of tan....?please help

[Cancer98765]

XD your videos are so much simpler and easier to understand/ remember than video lectures o.0....EVEN IF THEY ARE VIDEO LECTURES 0.0.....They are a different form or should i say art.

[ngyngxn]

looks easier and easier when i watch this video again and again

[eLeLiO23]

i have a question about this video. where is sec^2 when you derive (x^2+8x)^1/2 ??? thanks for all the videos. they are helping me on my summer class! :)

[Matthew2400]

I don't get how you can just suddenly know that the derivative of x^2 is 2x, of 8x is 8, of sin is cos, or tan is sec^2 and so on. How do you know all of that?

[MrWestudent]

You explained this so well.  Thanks so much!

[xtremdansio]

hey in the last part on the 2x+8 dont the 8 cancels out? since you are getting the derivative of the inside part?

[brandibomb]

why did I not know about you the whole time taking my calculus class?! it would've helped me soooo much.

[jioklp]

You make this looks so easy , its ridiculous

I am scared that maybe I understood it too well !

Thanks a lot bro !

[elizabethboeymusic]

you explain it better than my teacher!!!!!!!!



[Turquesai]

Wow. This example is a real test of your chain rule skills. I passed, thanks :)

[mikeybruises]

Thank you very much. You explained it far better than my professor did. You're a big help.

[dajakesta1234]

patrick do u hav any trig identities with the derivatives and stuff?

[Jendezi]

Wow, that was so easy... did not expect to understand this after just a few minutes. My TA takes 90 minutes to get flustered and confuse himself, as well as his students (at Columbia University!), and Patrick just clearly and calmly put it in plain English. I thought about dropping the class because maybe I wasn't intelligent enough to understand derivatives, etc., but this stuff is so easy! Thanks, Patrick!

[FSTK]

God-damn, I was stressin' about Chain rules until now. You going to be a teacher someday?

[patrickJMT]

@FSTK i used to teach and stopped. education is now viewed like any other consumer product: i pay, you give. i felt like i had to peddle grades in some cases. i just can not be a part of it. plus, teachers make cruddy pay and get no respect; no fun there.

[FSTK]

@patrickJMT

I know what you mean. I'm going to be a teacher someday and I'm going to work hard to change the system. Just because the system sucks doesn't mean the students should suffer as a result, I gathered. Pay sucks, but I'm not in it for the pay. As silly as it sounds/looks.

Fuck man, you're a fucking inspiration. I've no other words to describe how helpful you are to me in calculus right now that such a man couldn't stay and help change the system. We need more people like you.

[patrickJMT]

@FSTK good luck with the teaching! i hope it goes well for you. i feel like the videos ARE changing the system. i get a couple millions views a month; i could not have had that sort of impact will standing in the class room. the internet is helping to fuel the great democratization of education and i want to be part of it : )

[dajakesta1234]

@patrickJMT then how do u earn ur money???

[romacster]

In a very simple way you cleared up my confusion with the chain rule. Thank you kindly @patrickJMT

[Ladarzak]

@patrickJMT I hope things are going great for you. Your explanations are wonderful. My time in my calc class has been wasted. I get more out of 1 hour with your videos than 10 hours in that class - and no pain!

[jamesbond847]

@patrickJMT it also depends on the student you are teaching not all students show disrepect



[jacquesth2]

@FSTK lol this guy could do more than teach with his knowledge......i mean he could and would be great at it

[LAnonHubbard]

nice one

[Soupfinger2]

Does anyone else find this stuff to be 1000 times simpler than algebra II or trig? I had low Cs in those classes in high school and I'm doing great in Cal

[jvrcb17]

awesome!!!! this is sooo much easier than what my professor explains it. he decided it would be easier to start on the inside and substitute every little part with a variable. then he would have us substitute it back out. this is soooo much simpler, thanks for the help

[TheSolvie]

Sooo what happened to the 4th power at end?

[tupacsomething69]

not really complicated, just long...try this>>

> y=x^2sin(1/X)

if thts easy i got another

[konradoszczecin]

I'am from Poland but Maths is an international language, thanks to you :)

[G0DofRock]

i havent learned about derivatives too much but this problem seems very easy, just long. lol

[AliAlzaidi90]

you're way better at explaining calculus than algebra (pre-calculus)

[lilyjune1230]

how come you did not find the derivative of 2x+8 at the end?

[G0DofRock]

@lilyjune1230 i think its becuase 2x+8 does not have a power around it. That would be like taking the second derivative, which you dont want to do here. lol

[Outsider2000]

@nhuhoang92 its tangent ( sin x )

 it's not (tanx)(sinx)

[tacoburn11]

can somebody tell me why I absolutely love Calculus when everybody else hates it??? It absolutely intrigues me.

[patrickJMT]

@tacoburn11 not everyone hates it.

[gyuriFORhire]

@tacoburn11 yes i can. you are a fag, my friend.

[MustBeFiend]

haha i love how everyone is saying how this video is better than their professors and/or teachers. shows how bad public schooling really is..

[pratasov]

thank you so much, man. much better than khan academy(take it as a compliment)

[anticommon]

Honestly, my teacher fails to teach this material. Online videos such as these help me to actually understand. Thank you so much!

[kukupuffs93]

LORD HELP US ALL! lol

[MissKierie99]

Thank you :) Thank you so much.

[qwerty842]

@nhuhoang92

It's tan(x)

where x=sin(y)

where y=(x^2 +8x)^1/2

It's all compositions, there's no multiplying involved.

[bvlgary09]

my problems are more based on algebra rather than calculus. I get lost when I'm simplifying after differentiating stuff :(

[whyaminoob]

do you not times by 2 in the end cos it's 2x + 8??

[angellily0330]

um....like THANK YOU

[godss3]

dude you are awsome i like your video i am learning so much and this problem was fun i actually liked it xD thanks for making my day even more fun but i should play some games before my brain goes "co-co" xD

[bac1087]

Oh man, this problem was straight up fun.

[TehRimjobber]

Very clear and concise with no obfuscation at all. Thanks a lot! I can't say the same for the "Chain Rule and Parametric Equations" section in Thomas' Calculus 11th Edition. :X

[ezcondition]

what does the SAT have to do with anything? the greatest mind in the world da Vinci wasn't aloud formal study at all

[MasonMurdock]

Thank for you this great video. I do have a question though. The last part when we have take the derivative of (x^2+8) = (2x+8). When do we know when to drop the constant versus keeping the constant?

[kainmaro2010]

@MasonMurdock

it's the derivative of (x^2+8x) not (x^2+8) . whatch out.

[plustax50]

whi is it 2x+8 at the end? Shouldn't it be 2x and no 8?

[ezcondition]

@plustax50

you got him

[HoodSta93]

would this be on an AS maths, core 3 exam, im in the UK btw

[Yu2Kal]

@HoodSta93 chain rule is core 3 but the trig differentiation is core 4

[mayyoom]

YOU ARE AWESOME!

[maltamonument]

now thats confusing O.O

[mypinkdollprincess]

Ohmygosh! You made it so simple..honestly that problem scared the crap out of me when I first saw it! Thanks a ton!

[73337]

derivatives can get sooo fucking confusing

[niel423]

Thanks.

[guayreyes]

dude do you always get A's on your math exams haha xD great vid man. professors explain things in the most confusing ways your awesome thanks

[eddiesnumber1]

you make one mistake in calculus your azz is grass!

[xilllllix]

Question - why don't professors teach like this instead of teaching the longest method possible in order to confuse you? Aren't they suppose to help us learn?

[whatever1234312]

Thank You!! you explained this A LOT better than my professor.

[walleva69]

nice job helped me out!

[zain19]

chain rule is like a telescope

[DrPott3r]

what level maths is this??

[patrickJMT]

level 6

[popedarren]

@patrickJMT Yes. Level 6 at least. And you'll need your +5 Derivative pen or else your likely to be eaten by a grue.

DrPott3r, it's Calc 1 or Calc 2, but you won't need to learn it, just use the spell, "Solvus Maximus." And remember to spin flick your wand, not the opposite or else you'll get the integral.

[mysteriousnyc]

@DrPott3r Calculus 1

[Delrayjoe]

@DrPott3r survey of calculus

[MIT2004]

@DrPott3r actually this is level 1

[rashad9009]

thanks.. but hey! you forgot +c

[tbodowd]

That would be if you took the integral...right?

[patrickJMT]

of course

[matt8096]

this is differentiation not integration!!!!

[Catc1h22]

thank you

[limitbreaking]

Jesus Christ. You are a godsend. Thank you for your informative video -- I couldn't have felt confident in my chain rule without it, lol.

[tennistallie]

thank you this helps a lot. i am going to get a good grade on my test 2mro thanks to your videos!

[Futballaz]

wouldnt you find derivative of (2x + 8)

for last part .. woulnt it be .. (2x + 8) * 2

[mrH0TTINGER]

no, that would be finding the second derivative, which is not what he is doing here

[755hp]

It does make sense!! Man your the greatest!!

[georgespelvin]

Thanks for the helpful worked-through examples.

Should these minor changes be made?

line 2: "y" -> "y'" (missing prime after y)

line 2: ")]" -> "))]"

line 3: remove "]" at end (just before "->")

line 4: "]" missing at end

[ChoujiAkimichi]

depends on how anal your math teacher is about it.

I would take note and do those though.

[trunker420]

you are so clear and helpful THANK YOU!!!!

[unccfile13]

Thanks for posting great lesson here. One like it was on my test. I'm a little lost at the last step.

I don't understand why you multiply sec^2(cos((x^2+8x)^(.5)) times (.5(x^2+8)^-.5)*(2x+8).

Will you explain? Thanks Again.

[shananay]

unccfile13-

Which part is confusing? btw, you said sec^2(cos((x^2+8x)^(.5)) but its sec^2(SIN((x^2+8x)^(.5)) times (cos((x^2+8x)^(.5)). You're just taking the derivative each time. Derivative of tanx is sec^2(x), derivative of sinx is cosx, the derivative of (x)^(1/2) is (1/2)(x)^(-1/2) and the derivative of (x^2+8x) is (2x+8)

[theCrypticRain]

Thanks Shananay, the extra explanation made me understand the last step and the chain rule.

[VintageFlorist]

He multiplied sec^2(cos((x^2+8x^(.5)) by the next outside term, which is the ^(.5) power. He brought it down to the front and subtracted the power by one (that is the Power Rule). Then, he multiplied THAT by the next (the last) term - x^2+8x whose derivative is 2x+8.

I hope you understood what I tried to explain. :/ Sorry if I just confused you. :(

[ItachiIsGod]

THANK YOU SIR!

[marcusjhung]

Insane chain there, but the real pain starts when you try to combine and like terms and factor out stuff and put everything together. thats also when mistakes happen.

[mockstersz]

great stuff..it really help! thanks, dude.

[MetallicaFTW7]

wait wait wait... at the very last step... what happened to the secant squared before the cosine? you just have .... all that times cosine(x^2+8x) etc etc....

but what happens to the secant squared before that, the derivative of tangent?

[gangstajpimpin]

the derivative of tangent is sec^2, so thats where he got that from

[farlem1]

what did you get on your SAT at this ?

[patrickJMT]

me? i did not take any sat subject tests...

i did not take precal til i was 20

[cchapar]

rofl. as I did.

[maltiABZ]

It really did (: thankx a lot its more easier this way than i the way i have learnt it!

[ripsnorting13]

i get this, but multiplying it all out...forget it lol.

[silvver]

The second line should start with y' not with y

[ripsnorting13]

dy/dx

[popita2007]

@silvver you're right!! :)

[MasonMurdock]

@silvver thats something so nitpicky.. worry about that on your exam. If youre such an expert make of video on notation.

[gladeplugins]

You explained it in great detail, unlike my teacher! I wish you were teaching Calculus at my college. :D

[kirarei12]

You're incredible. Thank you so much.

[geminikhan23]

Well done!

It's a great example, helped me a lot.

Thank you.

[GoldAK47]

Fucking awesome.

[peaky010592]

good video, very clear and well explained, cheers.

[Thrawst]

I had to turn my paper sideways to do this, then when I got everything, I was just like: "damn, now what?" I really don't thing you can take it much further like you said, but damn.

oh, and thanks, this helped a bunch.

[hmongb2]

thank you so much for the videos, makes calculus a whole lot easier.

[jachina]

Thank you for the video, it will help me out a lot, since I have a test on derivative on thursday, :(

[patrickJMT]

dont be sad : ) it is only a test! : )

[Zernip]

LOL

[DeadlyArcAngel]

Great example, I always seem to get thrown off when some trig is thrown in there.

This is going in the notes!

[patrickJMT]

glad it helps!

[xylokid4]

yah it made sense

[clout90265]

perfect example...not too hard...just perfect. take the derivative of the baby, or whatever is inside the parethisese!!!

[patrickJMT]

ha, the baby! : )

glad u like it.

[MundusLitterae]

This is awesome! But I've done an easier example sin(tanx^2) where there is like no work. My answer is cos(tanx^2)(sec^2(x^2)(2x). I wonder if my professor accepts that there's no work because I know what I'm doing using your method lol =P But thanks!

[AznHavx90]

you rock!

[FarFromEquilibrium]

Patrick your videos a lot of help.I still get confused when getting these kinds of comps. even though it seems obvious once I see you expand it out.

[patrickJMT]

yes, they can be confusing! a couple of very smart people that i tutor were also having troubles with them, so i figured other people probably do too!

hope it helps though...