The Chain Rule... How? When? (NancyPi)

MIT grad shows how to use the chain rule to find the derivative and WHEN to use it. To skip ahead: 1) For how to use the CHAIN RULE or "OUTSIDE-INSIDE rule", skip to time 0:17. 1b) For how to know WHEN YOU NEED the chain rule, skip to 4:35. 2) For another example with the POWER RULE in the chain rule, skip to 7:05. 3) For a TRIG derivative chain rule example, skip to 9:33. 3b) For the formal chain rule FORMULA, skip to 11:36. PS) For a DOUBLE CHAIN RULE (or "repeated use of the chain rule") example, skip to 13:33. Nancy formerly of MathBFF explains the steps.
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1) The CHAIN RULE is one of the derivative rules. You need it to take the derivative when you have a function inside a function, or a "composite function". For ex, in the equation y = (3x + 1)^7, since the function 3x+1 is inside a larger, outer function, the power of 7, you'll need the chain rule to find the correct derivative. How do you use the chain rule? You can think of it as the "OUTSIDE-INSIDE" rule: take the DERIVATIVE of JUST the OUTSIDE function first, LEAVING THE INSIDE FUNCTION alone (unchanged), then MULTIPLY BY the DERIVATIVE of JUST the INSIDE function. Sometimes you might hear this expressed as: take the derivative of the outer function, "evaluated at the inner function", times the derivative of just the inner function. For our ex, first take the derivative of the outer function (the power of 7) to get 7*(3x + 1)^6 since the derivative "power rule" tells you to bring down the power to the front (as a constant or coefficient just multiplied in the front) and then decrease the power by 1, which leaves a power of 6. Notice that you leave the inside function the way it is and just rewrite it for now. Then you multiply by the derivative of just the inner function, 3x + 1. Since the derivative of 3x + 1 is just 3, the full derivative (dy/dx) is: 7*[(3x + 1)^6]*3, which is just 21(3x + 1)^6.
1b) HOW do you know WHEN TO USE the chain rule? If the original equation had just been x^7, there would be no need for the chain rule. It's when you have something more than just x inside that you should use the chain rule, such as (3x + 1)^7 or even (x^2 + 1)^7. Sometimes the chain rule may make no difference. For instance, if you have the function (x + 1)^7, taking the derivative of the inside function just gives you 1, so multiplying by that inside derivative of 1 will not change the overall answer. However, it can't hurt to use the chain rule anyway, so it's a good idea to get in the habit of using it so that you don't forget it when it really does make a difference.
2) Another chain POWER RULE example: To find the derivative of h(x) = (x^2 + 5x - 6)^9, use the same steps as above to first take the outside derivative and then multiply by the inside derivative. In this case, the derivative, dh/dx (or h'(x)) is equal to 9(x^2 + 5x - 6)^8 * (2x + 5). Using the chain rule with the power rule is sometimes called the "power chain rule".
3) TRIG EXAMPLE: the idea is the same as above even if you're using the chain rule to differentiate something like a trigonometric function. If you have anything more than just x inside the trig function, you'll need the chain rule to find the derivative. For the equation y = sin(x^2 - 3x), you first take the derivative of the outer function, just the sine function. Since the derivative of sine is cosine, the outside derivative (with the inside left unchanged) is cos(x^2 - 3x). Then, find the derivative of just the inside (of just the x^2 - 3x part), and multiply by that. Since the derivative of x^2 - 3x is 2x - 3, the full derivative answer is dy/dx = cos(x^2 - 3x)*(2x - 3).
3b) FORMULA: Although it's easier to think about the chain rule as the "outside-inside rule", if for any reason you have to use the formal chain rule formula, check out the two versions I show here. Both are based on the equation being a composition of functions, f(g(x)). The second version shown uses Liebniz notation. Either way, both show a component of the derivative that comes from the inside function, and it's important not to forget to multiply by this inside derivative factor if you want to get the right full derivative answer.
P.S.) DOUBLE CHAIN RULE: Sometimes you might have to use the chain rule more than once, known as "repeated use of the chain rule". In y = (1 + cos2x)^2, not only would you need to take the derivative of the outside power of 2, as well as multiply by the derivative of the inside function, 1 + cos2x, but after that you would ALSO then need to multiply by the derivative of the 2x inside cosine because that inside function was 1 + cos2x and not just 1 + cosx. This means you would use the chain rule twice. The idea is that you have to keep taking the derivative of the inner functions until you have reached every inner function that is more complicated than just "x".
For more calculus math videos, check out: nancypi.com

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[@Hugs_4_Bugs_]

Something a lot of students don't understand is that professors are not meant to teach you everything or hold your hand through a unit, just show you the concepts. It's your responsibility to practice outside of class. Back then when there was no KZbin, people had to just read the book or find a tutor. This generation has it pretty good imo. But yeah some teachers are pretty bad at what they do. But if you don't ever practice, you can't expect to get good at something. Just saying 'cause it's dumb when students blame the teacher for getting bad grades when they just play fortnite or CoD all day lol

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"time is the one thing you can never get back" i think to myself looking at all of my calculus homework

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lol I hope you're not saying this in Calculus one. Ha!! Wait till you get to Multivariable calc and differential equations (if you're a math major) to say that!

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I also know one that does that and it irritates me to the core. They should explain to us, not to shout at us.

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Teaching & learning do seem to lend themselves to one-on-one, student-led desire of learning. E.g. We're watching this video, and videos like it, precisely when we want to -- and can skip, rewind, choose another, walk away, etc. whenever we choose; and, we're most likely watching it because we're actually interested to learn, and prepared right now to take on the labour -- a labour of learning that, since we've chosen it, is actually enjoyable. Compare this to a classroom with 20+ students being taught by someone who likely is thinking (possibly dreading) their next class, and who may have just been teaching yet another class (all of which precisely when they're not really in the mood for it). I.e. Nancy, Saul, et al. are never under the stress of actively controlling a class (cf. Eddie Woo!); plus, even given Eddie Woo's case -- their videos are not classes, but discursive lectures with 0% threat of interruption or time pressure (something where they can just talk and explain, and which they can prepare for single-mindedly and reshoot and edit -- it's hard to edit real life etc.). In conclusion: this is not to defend teachers who scream at us, but just to say that a classroom with 20+ students, run according to a schedule that fits neither the student nor the teacher but just for logistical necessity, is about the worst environment you could possibly design for learning (or teaching).

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knowledge is the best candy

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It's been 3 years already since i watched Nancy's videos. Thanks a million times. You made me love math. graduation bells about to ring💖. Just had to come back and say thanks.

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Khan academy is full of smart people. Being an MIT math grad is a whole, notha, level

[@stiventson4464]

To me the khan academy explanation was way better than this, because they demonstrate the chan rule

[@elimarburger1659]

@@stiventson4464 please explain what the "chan rule" is

[@TerryJLaRue]

@@elimarburger1659 The "chan rule" is another method of finding derivatives invented by the famous Honolulu detective Charlie Chan.

[@laurendan20]

I would watch NancyPi's videos religiously throughout high school when I was taking Pre calc and Calc, and this one specific video is still on my recommended page everyday. I just graduated from high school this past June and I just wanted to say thank you so much Nancy. I've never been strong at math (I'm an English/history person) but your videos helped me out tremendously. I'm starting college in a month & majoring in education. I hope to help as many people as you do!

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she is basically the best person i have ever watched to teaching math, i cannot even begin to explain how high school never taught us these things and always made math feel so damn complicated like you have to be einstein in order to understand how the world works. And now i am suffering from teachers who cannot explain what limit means and complex it even more in their own way of vision, thank god for nancy helping me finally get somewhere decent around calculus.

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@@dardoraercument2887 it really helps that she doesn't over complicate it and clearly understands how much the student knows/doesn't know.

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I am currently in Calc III after taking more than 10 years off after undergrad, and videos like this are a life-saver. I've learned the material, but it's been so long... Great refresher!

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one thing i love about Nancy, is that she simplified math esp calculus i am a kenyan who feared math but now i can tacle hard looking math i now love maths thanx nancy

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right like even teh basic algebra that some of us might forget

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I didn't know Megan Fox was teaching math online, didn't see that one coming.

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I didn't know she could write backwards

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@@millenniumdecade4628 she doesn't, the video is mirrored with respect to the y axis

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@@rikimbili No, she is actually right handed, so it's even more impressive how she is able to write backwards with her left (not main) hand.

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@@renevos5822 Being able to write backwards and upside down is a skill you can learn. I have to write for people facing opposite me, so I've learned to write backwards and in the opposite direction to avoid wasting time constantly pulling the paper towards and flipping it around. It's a bit brain melting at first, but with practice it becomes as natural as regular writing.

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[@kookiebobateabubblesjeon8079]

I graduated high school early in junior year this year and I was worried that going into college early would be a real struggle. Especially with Calculus. But then, I searched for how to find a derivative and found Nancy. Thank you so much for simplifying it so well! I'm able to understand the concepts even more now!

[@Audioholics]

Dear Nancy, your presentation is inspiring me to do advanced math again despite I've been away from it for nearly 2 decades. Thank you for doing these videos. Simply superb! You should definitely be monetizing this channel more. You deserve it!

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i don't think i've ever left a youtube comment before but for you, i just have to! i have such a difficult time following most math instruction because i NEED the constant reminders of which rules are being applied where, even for simple things like simplification. You explain things so well, so thoroughly, and are genuinely the best math teacher i've ever had! i just wish you had more videos!! thank you a million times over

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Hi Ari. I totally understand what you are saying. I failed Calc-1 2 times!! Therefore, I have now a channel to teach math. I am just applying there what you are saying here.

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One attribute that separates your videos and others is the reassurance of rules. You touch base on the basics before moving ahead too quickly. You make the answer make sense creating more confidence down the road!

[@ElonMusk-FanZone]

My teacher can waste hours on this with her crappy lessons. I learned it from you in 5 minutes. Thanks!

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Butcha gotta focus

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Really, 5 minutes?

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@@beautifulstrangers3748 I always speed up videos

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Congratulations on becoming the richest man

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I will come and take the richest man record.

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my exam is tomorrow..Where the hell i have been this long..finally kinda liking maths..what took u so long Nancy. You are really a life saver. THANKS FROM HEART

[@chloepatti9734]

Okay, so my two friends and I were told to take notes on this video as an assignment. We all agreed that you are one of the best math video producers we've ever seen. We also agreed that you have a chill vibe, like to the point where we could make flower crowns together and talk about life with you for hours. Seriously, this is a fantastic video! :)

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I think I'm INLOVE with this GIRL. She is so calm and her voice is so sweet to hear. I learned a lot from her discussion. I love you...

[@Deejay_Ezeugo]

Honestly, you are one of the best tutors ive ever met. You dont know how to solve math problems but you know how to slay it. Kudos

[@erin2150]

Thank you so much! I've been struggling with derivatives for some time now and really appreciate you doing this

[@R94jax]

You definitely had a huge part in the development of my lov.... appreciation for math. I was going to say love, but Multivariable Calculus had definitely made me cry more times than I care to share. Thanks again!:)

[@johncoleman4265]

NANCY YOU ARE THE GOAT, literally just saved me when it comes to differential calculus. Calc text books /professors notes can over explain leading to confusion, you make it easy to digest the concepts. Thank you!!!

[@Ppylv]

I really appreciate your teaching style which should be universal but isn't always. You teach with the definition of it, by teaching slowly and clearly so we can learn, expecting that the people watching don't understand it previous to this. Some teachers in person and online ramble on expecting their students to somehow magically catch on.

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Hi Nancy, I just wanted to say that I appreciate your videos and your efforts so much. They mean so much to me. You are a cornerstone of my success. Thank you for helping me so much!

[@emmasprague5465]

came for calculus help, left with an existential crisis

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you are a life changer. Nancy, you really developed my interest for calculus. You are more outstanding than all my calculus teachers i ever had. thank you so much

[@hichica1462]

Wow, you're probably the most helpful math youtube I've ever seen! All of your videos are so helpful and you're great and explaining! Keep it up and thank you!!

[@shamineywalsh-dupuis6520]

I thank God the day I came across your channel. I love how you make these things look so easy. Thank you Nancy ❤️. You are a truly amazing teacher

[@PriAndFiction]

Thank you! These videos are really getting me through AP Calc.

[@adrianbenson2570]

You are a LIFESAVER. I'm so glad I'm just now finding your channel my sophomore year of high school. This should get me through all sorts of calculus. Thank you so much

[@abloodorange5233]

Exam tomorrow... My big brain decided I didn't need to tune in for class for the past 2 weeks, you just explained it to me in 16 minutes! Life saver!

Clear and concise explanations with diverse and appropriate examples using excellent instructional technique. Thank you for making this video.

[@orangethew]

damnn no likes?

[@meganwing3736]

This helps so much during online learning because of COVID-19 :)

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Yah really i am a maths lover so could you be my friend then send me your mail

[@meganwing3736]

Saad Zia What?

[@caitlin.7]

@@meganwing3736 ahahahaha

[@Sanguifier]

Just thought you should know, even 4 years later this video is still helping teach calculus! Your method of explaining stuck way better, my professor was using g's and u's and crisscrossing arrows, and well, it did not click lol!

[@carlosdanielgt]

Never understood the chain rule so clearly as I did after watching this video. So far the best chain rule tutorial I have found.

[@mramakrishnaraju]

Hi Nancy, i have most of your videos and i really appreciate the efforts you are putting in. Nancy along with these tutorials if you can videos on how to multiply 3 digit, 4 digit, ... numbers. Divisions, square and cube roots short cuts etc will be a great help for others and your followers.

[@randomthingz8010]

omg she's so calm and sweet. I wish my school teachers were like her😭

[@erfanahmadi2569]

Thank you so much. This video really helped me how to use chain rule. I've been taught chain rule in a really complicated way; however, this one was just so simple and clear that anyone can understand.

[@lyndseystrait1513]

👏Great video & explanation of the chain rule! Comparing instances when you do and don't use the chain rule, the use of different colors, & showing both the Lebiniz and Newton versions of the chain rule formula was very helpful! I now understand the Lebiniz format for finding derivatives & thank you for the heads up on double chain rule instances! Nancy's voice was soft yet engaging.

[@olivernardbagay318]

Its difficult for me to graduate in engineering without the help of nancy's math videos. Infinite gratitude!

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what year are you at?

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@@wuwu1598 1st what about you?

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@@unknowngeezer5029same

[@devayanisunil1466]

Can we take a moment to appreciate the title of the video? Like that was exactly my state of mind while clicking this vid

[@peacemaker2427]

I used your videos on limits and derivatives to teach myself what I couldn't learn in the past month and a half. Only watched three of your videos before my test today, and I got 6/8 questions right. It doesn't seem like I did very well, but before today, I would've gotten 0 questions correct. Thank you.

[@djgulston]

Watching these videos takes me back to university in 2016 and 2017 when I was doing calculus for the first time. Miss those days! ❤

[@alanduncan3710]

Calculus taught me the most important lesson I ever learned in school. Don't take a class at 7am five days a week. On the bright side though, I always got a good parking spot!

[@anaidceniceroscruz6752]

I really needed this! Thank you! 💖

[@gauribharadwaj9120]

I am a highschool student but in my class, we have not yet reached calculus. I had been trying really hard to understand all these concepts and watched a hundred videos but could not understand anything until I found your channel. It really helped, thanks a lot!

[@sceneryj]

Of all the reasons to watch this video, the tone with which “its trig… don’t get too excited” is delivered is top.

[@UwU-dx5hu]

"time is the one thing you can never get back" thank you

[@starlightjournal]

Hi, Nancy! I just wanted to thank you for this video! I've been so confused in my calculus class, and I didn't do that good on our first test or any of the recitation quizzes. But you are making learning calculus so easy so far! And I really love how gentle you are and explain things slowly and patiently. Thanks a lot again! 😊

[@kayleebaker6889]

LIFE SAVER!! I have not understood the chain rule up until this point, thank you so much!!

[@jamjellies]

Thank you NancyPi, your approach to teaching is so amazing. This video has been very helpful. I will definitely view your other videos. Thanks again for your help. You are a superb Math instructor.

[@ashishsahota4751]

You explain better than all of the teacher in my school's math department

[@alihashimi9017]

Swear, wish you were my teacher. You’re relatable and know how to teach.

[@baobaomeow]

I have a final exam for my Saturday math school, and I've been just watching your videos to review. I really want to thank you from my heart. :)

[@justinpennino2386]

All I can say about you as a KZbinr is thank you. You've assisted me for years through academia, and I prefer to view your content because you bring all the necessary study material here, maximizing viewer's time efficiency. And to the other viewers - If you plan to be an academic researcher in STEM, you need to know mathematics. Mathematics is the language of science. I'm practicing to be an organic chemist, and things like quantum mechanics using Schrodinger's wave functions will be introduced to you down the line (assuming no new theories develop). Scientists need to understand mathematics, chemistry, biology, and physics, and be able to apply them mechanically to be successful. These concepts are just the beginning of your journey.

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How is it going now bro ?

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Dam Nancy! Way to go hella deep in the first 15 seconds! lol

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thats what he said

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not he rather it is she.

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Ma'am I'm from India you explain the thing in such a good way with clearing the concept like no other Thank you so much ❤❤

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30% audience is Indian!!

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@@ashwingoud5666 Actually a plot twist

[@Garold_Taha]

A simple thank you or a long post cannot express how appreciative I am in this very moment. Thank you.

[@DaquanMHall]

This was amazing. Thank you! People like you make math approachable for everyone.

[@tebohomaema3851]

Wow, it's my first year doing Calculus, I loved the way you have explained the Chain rule, you are so clear despite time which you released this video, it's still working for us beginners 🙂🎉

[@briank831]

Now calculus is so simple, contrary to what i was fed with when i joined campus, thanks to Nancy.

[@singhsingh3763]

Thanx nancy Gotta feels like I am again studying maths you really helped me a lot I was a maths nerd one year before. your videos are like life saver for me so I think to give you big thanx just bcoz of you I did best in my final exam of college and now I got a job in Microsoft in California as a data scientist you make my life

[@canyoustayupallnightstudyt6535]

Such a talented teacher. The only teacher ever I've found online/offline that made chain rule this simple to understand.

[@canyoustayupallnightstudyt6535]

And you're very pretty

[@ElonMusk-FanZone]

Thx so much! Teacher is teaching for hours and hours and I can't focus because it takes tooo long. This short video gave me more information than her!

[@kristindoja44]

"The Chain Rule is your friend" lol the way she said it was so so so sweet

[@CarminesRCTipsandTricks]

After 3 Videos of yours, I feel like I've learned MORE than in an entire Semester!! I'm also staggered by how you write SO well, BACKWARDS!! 😲

[@moraxstudios5314]

Thank you so much for your phenomenal impact Nancy

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Damn these videos are a whole lot better than my precalc class. 1) She talks slowly enough for my dumb brain to understand 2) She's a whole lot cuter than my 50 y/o teacher

[@mmmmnda]

Yikes... lets not reduce people to appearances... 😬

[@kaiwilson5218]

@@mmmmnda To be fair their first point was that she explained things slowly and well so it's not really reducing.. more supplementary?

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Haha you said it

[@AmigosOfTheTrail]

What's more trippy than the chain rule? How she writes backwards perfectly. Anyone else wonder how she does that?

[@jaden7983]

She did mirroring

[@colonelarmfeldt8572]

Wrote it normally (or backwards/mirrored from our perspective) and then mirrored the video.

[@mad_squidward]

Thank you for saving my college math life i'm on my last math class which is dreaded business calculus I don't know what I would do without your vids.

[@agnivachakraborty719]

This explanation on Chain Rule is the best on KZbin . Thank you so much

[@CrazyAlaskanCoDShot]

she sounded so sweet and kind at the end that I had to hit the like and subscribe button XD

[@sansamman4619]

this is stuff i know for like 2 years now, i am just watching to hear her voice it so soothing :)

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ikr

[@PimpMatt0]

Same but good refresher. Surprised how much easier it is to follow now than it was then.

[@mariosilva000]

It is fantastic when persons like you, have a pleasure to help others, the sounds of your voice sounds like music, has charm to teaching, thanks a lot!.

[@anthony-jt2mv]

I just understood a concept that has given me headaches for 3 days... the way I feel can't even described in words.Thank you so much

[@Hope4NewDay1]

"time is the one thing you can never get back" Well this shit just got real.

[@stingaling]

"It's Trig. Don't get too excited" !!!

[@pizzalover1790]

thank you for explaining the most BASIC concept that i didn't have down before starting more complex problems... i had no idea the chain rule could be used for trig

[@aboudoucara4293]

The way she is chill to solve this is unbelievable

[@moekiyam2673]

8:51... why arent we finding the derivative of the function again since we arent done? Shouldnt we be multiplying by 2?

[@nagendranamarnath9862]

Happy teachers day nancy

[@twogasp]

Thank you, Nancy. You rock. You helped me get a 100% on the Chain Rule quiz. Q1: y = cos(ln(x)) Ans: y' = -sin(ln(x)) * 1/x

[@mastrake]

Thank you! That was a brilliant and simple explanation. "Outside/Inside Rule" resonates.

[@hardcoregary8427]

Do you think you can do a video on differential equations? Just Ordinary differential equations.

[@iPro95]

cope with heartbreak with derivatives yes

[@DecafMathASMR]

Levi lol

[@CrazyAlaskanCoDShot]

lol

[@namr1174]

Lol

[@nielstielemans152]

Man I love it when it finally all makes sence!

[@kimiko495]

This is such a clear and easy to follow explanation of chain rule, thanks so much for sharing, I wish you were still making more videos!

[@neural8885]

Good morning from Australia. Thank you for waking me up via youtube notification lol

[@michaelperkin89]

Oh Nancy, You ae heaven sent

[@sax47345]

Awesome! You rock! The most clear explanation I have seen, and now have a clear concept of the chain rule

[@juliadavis9993]

Your videos are just fantastic. You have such a gift of communicating what appears complicated clearly. Phew!! Julia x

[@sirajtayyabkhan6820]

When God merges intelligence and beauty together, we see Nancy teaching us. Thank you Nancy and thank you God.❤😂 I know it sounds crazy.

[@aran_237]

You have saved my ass so many times, thank you

[@markkennedy9767]

Nancy's mock-confused facial expressions are great for teaching. She understands how to get down to the student's level and engage them.

[@thesubcontinent3708]

Thank you @NanciPi for providing such an informative video. Your kindness and the ease with which you taught this chain rule is alluring. Love and respect from india

[@malathibhat7373]

You teach great Nancy! Wish you were my maths teacher

[@nikkiw6788]

5:36 heard a tummy growl

[@rusucatalin7742]

Dude, you ok?

[@marnisantonites7013]

hahahahahahaha

[@MrFamousguy13]

How the hell did you catch that LOLOL

[@amardeepbanga9383]

supersonic ears

[@vangarde0348]

Bruh

[@sellakc]

I absolutely LOVE these videos. You are a fantastic teacher!!!

[@dhrubajyotisaha8539]

I literally took months to understand the chain rule.. But after this video, I clearly understood the usage of the chain rule. Thanks for your help