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.

[dyer]

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

[PrettyMuchPhysics]

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

[Meth]

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

[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!

[completeandunabridged.]

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

[MrCigarro]

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

[rockyjoe]

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

[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)

[tomatrix]

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!

[MarioFanGamer]

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).

[adamhrankowski]

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. 😎

[everettmeekins]

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

[andreapaps]

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

[JamalAhmadMalik]

Precisely what I wanted to learn today!

[fromblonmenchaves]

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

[AlmightyMatthew]

I skipped to the end just to hear the story again

[rogerkearns]

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

[hamez]

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

[zorak]

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!