Chain rule proof | Derivative rules | AP Calculus AB | Khan Academy

Here we use the formal properties of continuity and differentiability to see why the chain rule is true.

AP Calculus AB on Khan Academy: Bill Scott uses Khan Academy to teach AP Calculus at Phillips Academy in Andover, Massachusetts, and heÕs part of the teaching team that helped develop Khan AcademyÕs AP lessons. Phillips Academy was one of the first schools to teach AP nearly 60 years ago.

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[gothicknight]

Thank you. I always love to know the proofs and not just simply memorizing it.

[aditt.]

The first ten seconds of this video was the most accurate description of the chain rule that I've ever heard of.

[papad]

This is not a proof. The proof of this requires either non-standard analysis ( which you seem to be assuming to hold true in this case) or the FTC.

[arielfuxman]

This just shows how useful the d/dy notation is.

[yotsubakoiwai]

y needs to be differentiable at u(x), not x

[ceryan]

Your proof of the Chain Rule is technically incorrect. For example, what if delta u is 0 (i.e., u is a constant function of x)? In order for your argument to work, you must assume delta u is not 0, since you're dividing by delta u. Then you would have to consider the case where du/dx=0 separately.

[MrZeroSugar]

Thanks so much! I just learned the chain rule at school, and was a bit disappointed that we didn't learn a proof.. and here it is! :)

[KBineetPrasadPatro]

Thanks sir, I was founded from a long time, finally I got it . So, double thanks ☺️😊👍👌💐🎂

[Hobbit]

I came up with this alternative proof:
We know that: df=f'(t)*dt
Now, if t itself a function of another variable x then we have that: t=t(x)=g(x). Also dt=dg (that is, an infinitesimal change in t results in an infinitesimal change in g)
if we plug this into the first equation we have that: df=f'(g(x))*dg
Then we divide both sides by dx:
Is this correct???

[zboltable]

Excellent.  So by the same logic you could turn delta u into delta z.  But, the reason that doesn't work is you have no way of calculating delta z with respect to x.  The reason this works is because you can calculate delta u with respect to x.

[barthennin]

Great proof up to the point he says, "according to a previous video, this is true" (Shaking my head)

[qualquan]

Just finding complex F'(x) by using the very useful U substitution of the "inner" x terms (function) and then finding dU by dU/dx to express everything in terms of x as a diff. coeff. Is a proof needed to justify substitution?

[notsodope]

I don't understand why is it important to consider the case when u(x) is constant function coz usually its not used. Idk

[aakashkarajgikar]

5:37 I would like an actual proof please. Is dy/dx a fraction? My calc 1 teacher/professor I have is telling me it isn't a fraction, but I am hearing wrong. So I am pretty confused.

[zainkhalid]

this video was a helpful video for me thank you.
Question:
(where the chain rule is use, how do we know where we use chain rule, chain rule daily life example)
Thank you

[black_jack_meghav]

Dear sir, isn't this proof incomplete? ∆u can be zero even if ∆x isn't. This is what the textbook says.

[littleuniverse]

Can you do a proof by the definition of derivative ?

[nafrost]

4:58 I see here that you are replacing one limit here with another. Is there a name for doing that? Also what are thye conditions, that allow you to replace the limit as x -> c1, to the limit as u -> c2?

[asyraafafandi]

what is the title of the previous video???

[martyspandex]

At last! I've been waiting for a decent proof for the chain rule for ages. Good stuff.