Proof of the Chain Rule

In this video, I provide a neat proof of the chain rule, and I also explain why I call it the Chen Lu. Enjoy!

[@AndrewDotsonvideos]

If you think of the partial derivative as the component of the gradient vector, chain rule just pops out by definition of how covariant vector components transform which is pretty neat. Not sure if it also works for chen lu.

[@drpeyam]

Chen Lu, not Chen Lou 😅

[@AndDiracisHisProphet]

Chen Lew?

[@Demki]

Chen Lieu

[@koenth2359]

en.m.wikipedia.org/wiki/Chen_Lu_(figure_skater)

[@lucasfrykman5889]

@@koenth2359 Imagine every calling her chain rule now lmao. She will be so confused.

[@jessehammer123]

Chen Liu.

[@dyer308]

Dr peyam i took complex analysis this semester because you motivated me so much to take it all last year!

[@completeandunabridged.4606]

I have only recently started watching your videos. I love them.

[@wankar0388]

Thanks Dr. Tigre Peyam. You are the best!!!!!!

[@rockyjoe3817]

Time to understand the rule that the teachers made us memorise it !!!

[@MrCigarro50]

Clarísimo, Fantástico...Gracias, Dr. Peyam.

[@Meth24096]

Simple, efficace ; merci docteur, vous mériteriez vraiment plus de visibilité sur la plateforme ! (PS : Chêne Lou ça marche aussi)

[@quantumsoul3495]

Nn c'est tchene lou

[@dougr.2398]

Ou peut-être chien-loup

[@SisypheanRoller]

This is one of my favourite episodes of the Dr Peyam show. I feel the Chen Lu coursing through my veins now 😁😁 Time for the HD Chen Lu!

[@PrettyMuchPhysics]

Chen Lu *does* sound powerful... 🤔 Maybe I'm gonna use it in a future video.. :D

[@JamalAhmadMalik]

Precisely what I wanted to learn today!

[@Demki]

I've seen a different proof that defines the two functions : given y_0=g(x_0) (and calculating the derivative of f composed with g at x_0, with g differentiable at x_0 and f differentiable at y_0) F(y)={f'(y_0) when y=y_0, (f(y)-f(y_0))/(y-y_0) otherwise. G(x)={g'(x_0) when x=x_0, (g(x)-g(x_0))/(x-x_0) otherwise. Then shows that (f(g(x))-f(g(x_0)))/(x-x_0) = F(g(x)) G(x) since both F, g and G are continuous at the appropriate points (which is because f and g are differentiable at y_0 and x_0 respectively), when taking the limit we get F(g(x_0)) G(x_0), which is by definition f'(g(x_0)) g'(x_0)

[@andreapaps]

Was your lecturer Chinese because in Mandarin （律 =lu) means rule and I'm assuming 'Chen' is chain pronounced with an accent?

[@drpeyam]

Ooooh I had no idea that Lu literally means rule!!!

[@MarioFanGamer659]

When I had to prove the chain rule, I never really understood why it worked (it doesn't help that the schoolbook I had has got a confusing proof). I knew that I needed a second variable (k in that case) which is dependent on h but I never came into me where that k came from. The punchline for proof of the chain rule is to define k as v(x + h) - v(x) (numerator of the inner differential quotient) and make the outer limit dependent on k instead of h. Due to the definition, if h -> 0 then k -> 0 too because v(x + 0) - v(x) = v(x) - v(x) = 0. That way, you can define as u(v(x + h)) - u(v(x)) as u(v(x) + k) - (u(v(x)) as shown in this video and derive the outer function. Of course, your proof goes even more technical but the most important step in the chain rule is to define k as v(x + h) - v(x).

[@tomatrix7525]

Peyam, 100k is getting so close for you. I have no idea how you didn’t reach it years ago, but atleast your still our little secret lol!

[@drpeyam]

Awwww thanks so much!

[@nestorv7627]

That was beautiful

[@adamhrankowski1298]

Hey Dr. P. New subscriber. Love your vids. One suggestion on this one is that the board doesn't fill the screen. Makes it harder to follow on a phone screen. No probs. I can grab the tablet if needed. 😎

[@rajendramisir3530]

Same here Adam. Difficult to read on my phone. Beautiful proof though.

[@dougr.2398]

Rotate sideways :)

[@everettmeekins1582]

At 13:52 I am a tiny bit confused. So you are taking the limit as h->0 of the error term of h squiggle (which has h in the def and would to zero as h goes to zero. Though with that logic, wouldn’t you have had to move the limit inside of the error function? And to do that, we would have to prove the the error function is continuous right? I just don’t remember if that was done or it is was super easy to see why

[@pauloedmachado9137]

Everett Meekins when I had to do this proof in my Real Analysis test, yes, we had to prove the continuity of the function. It's a really long and dull proof

[@everettmeekins1582]

paulo ed machado oh okay. I just started taking real analysis the semester (only have had 2 classes). But rip tho

[@-_-rain5432]

Awesome

[@hamez1300]

6:52 are we allowed to multiple out by h squiggle if it's zero? Would the zero denominator issue?

[@AlmightyMatthew]

I skipped to the end just to hear the story again

[@Gold161803]

So what actually changed when you replaced f(x+h)-f(x) with h squiggle? You just substituted it back later... Sorry if I'm missing something obvious

[@shuddhoshawttoroy6257]

Please proof the theorem of chain rule for integral calculus

[@rogerkearns8094]

_Use the Chen Lu!!!_ ...and don't forget to wash your hands.

[@dougr.2398]

ONE potential problem..... we agreed Sigma is some junk..... but if it depends on h or h « squiggle »(I prefer to call a tilde a tilde, unless it is a Waltzing MaTilde) suppose Sigma blows up? Can we show sigma is stationary (constant) or only increases in such a way that it’s product with f(x+h) - f(x) still goes to zero?

[@harpleblues]

In high school chem I heard my chem teacher saying heaven god knows number when talking about the number of atoms in the mass of an element. I was in complete agreement. How could you know the number of atoms in a given mass? Once I got my ears on right I heard him say Avogadro’s number.

[@orphixigl1476]

Why sigma -> 0 when h ->0? Sigma isn't a continuous function when f(x+h) - f(x) = 0

[@zorak0044]

Leithold says the function F in delta u terms, might be continuos at zero.. but Why? Derivation`s condition? I am a engineer , two weeks stuck wthis! Greetings from Perù!

[@dgrandlapinblanc]

You are so crazy Blackpenredpen and you ! I understand now. My best friend ! It's good all that the morning. Thanks.

[@xiding3709]

Chen lu can also be the Latin expression of a Chinese girl’s name。。。。。。

[@fromblonmenchaves6161]

Why is there a junk term in the g'(y) ?

[@jesuisravi]

Is there an easier way?

[@drpeyam]

Nope

[@lucasfrykman5889]

I have a faster proof. It involves what you already did with factorizing out f'(x) out of d/dx g(f(x). the 2 terms you are left with is (g(f(x+h)-g(f(x))/f(x+h)-f(x) * f'(x). I will focus on the first term. substitute f(x+h)-f(x) =u and f(x)= k You'll be left with lim as u--> 0 of (g(k+u) - g(k))/u * f'(x) Which is just g'(k) * f'(x) = g'(f(x) * f'(x). Bada bing bada boom QED baby.

[@Flanlaina]

Use the Chen Lu!!!!

[@slutskystheorem15912]

Is it Lou or Lu?

[@cicciobombo7496]

Loo

[@drpeyam]

Chen Lu!!! 😄

[@drpeyam]

Haha, how embarrassing, I wrote Chen Lou 😅

[@helloitsme7553]

Simplest proof: dy/dx =dy/dx * du/du = dy/du * du/dx

[@blackpenredpen]

HelloItsMe Love it!

[@dougr.2398]

Doesn’t that circumvent the limiting process as part of the definition of the differential? So.... pretending the numerators and denominators of differentials are separable is quackery!!!! (EverybodyDUCK!!!)

[@sarunyapongkeangsarikit2162]

I am very interesting but. I am not see anything because it too small

[@silly_dong]

I was an awsome man, then I was

[@adamhrankowski1298]

This seems like non-standard analysis, with the sigma term being an infinitesimal.

[@hunter-nj9kp]

Quit ur job kid im smarter