The Multi-Variable Chain Rule: Derivatives of Compositions

Рет қаралды 89,205

Күн бұрын

Suppose that f(x,y) depends on two variables but that the x(t) and y(t) are themselves both functions of t. Then f(x(t), y(t)) is a composition of functions and the derivative of f with respect to t is computed via the multi-variable Chain Rule. In fact, this scenario is one of many different generalizations of the single variable chain rule. We can use arrow (or dependency) diagrams to illustrate the relationships between a bunch of multivariable functions and for each situation write down a chain rule that gives the slope. The idea in the example given here is that a small change in t results in a small change in both x and y which in turn result in a small change in f.
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Пікірлер: 85

@lolololol632 3 жыл бұрын

You literally saved me hours of work because your explanation is so intuitive and straightforward, and in such a small amount of time. I wish you the best in future.

@DrTrefor 3 жыл бұрын

So glad it helped!

@balakrishnakumar1588 4 жыл бұрын

Why he isn't famous yet ? Great explanation sir.

@subhadeeppal917 4 жыл бұрын

Really I just love your videos. 🤩

@NuclearPhysix 3 жыл бұрын

He is famous. You may need to remember that not many people like or do mathematics, at least on KZfaq, you will not expect his followers to be the same number as some funny cat videos posts viewed by anybody else. He hi up there with MIT Herb Gross in my opinion

@AbhishekKumar-jg7gq 3 жыл бұрын

Because he has a difficult accent which non native speaker can't understand

@aashsyed1277 2 жыл бұрын

@@AbhishekKumar-jg7gq no.

@tevinabeysekera6038 Жыл бұрын

@@AbhishekKumar-jg7gq ?

@Lostwolf16 4 жыл бұрын

Content and quality of video is masterclass, hope to see a rise in quantity of other concepts as well. Thank you

@vaisuliafu3342 4 жыл бұрын

these videos are so good... i hope you are dreaming big enough with your channel. This is the future of math education

@daumel98 Күн бұрын

I love you. Visualizing a dependency graph makes this so easy! I finally understand backpropagation in neural networks

@hu900 2 жыл бұрын

I'm so happy that I found this, he explains it realy well. At 3.20 "If you only change w" should be "If you only change x". No problem though, it is clear anyway. Thank you so much for this video!

@PD-vt9fe 3 жыл бұрын

Great video as always. It helps me a lot to rebuild my math background. Keep it up!

@belemaayonoadu9219 2 жыл бұрын

Brilliant! I can't believe how you made this concept extremely easy!

@immortalkarin 2 жыл бұрын

Great videos man, you really have a very charismatic way of explaining math.

@Chris-ng9zi 2 жыл бұрын

You are an excellent teacher!!! Keep up the good work.

@honkhonk8009 Ай бұрын

Love the video bro. Im a CS major and was struggling to figure out how chain rule worked for backrpopagation. The chain rule seemed easy enough on textbooks but i never understood what the dependency chart thing was, or how it would be extended to other examples This just explained it perfectly

@salvatoregiordano9050 Жыл бұрын

Excellent video and explanation sir! I really appreciate this. Do you have any idea where I can find a video that proves the formula for the chain rule you described? Thank you so much!

@gokulakrishnansaravanan7475 3 жыл бұрын

u are really doing great sir. giving the real insight to mathematics .

@EfremTesfaye-cz7tl 3 ай бұрын

I get this video after a period of time but i regret about the time before. really i appreciate you continue don,t stop here .

@mimstudent8662 3 жыл бұрын

I am from Bangladesh.U teaching method is very good.I can get easily understand.I saw many videos but u videos really well done.Thank u so much.Sir

@sahilafroj277 3 жыл бұрын

Beautifully explained sir. Love you.❤️

@isakwatz11 3 жыл бұрын

Impressively simple yet effective expanation!

@amansingh-ww2qc 3 жыл бұрын

Best maths teacher ever

@xiu8548 4 жыл бұрын

you r not only more handsome than my professor, but also give GREAT explaination! ! My professor gave lectures in 2 hours, you explain it in few mins.

@aravindhvijayanandan3010 2 ай бұрын

Great exposition of a difficult concept, Dr. Bazett Subs from India

@crimfan 4 жыл бұрын

A very nice explanation of the total derivative.

@jitendermeena9480 3 жыл бұрын

I loved the tree diagram. Great job sir 👍🏻

@jankowalski8102 2 жыл бұрын

Great content! Greetings from Poland!

@billhopkins8330 4 жыл бұрын

Great explanation!!

@joyghosh3858 3 жыл бұрын

This is how math should be taught.. Just superb.

@DrTrefor 3 жыл бұрын

Thank you!!

@yosafendrafendra7960 3 жыл бұрын

Hello Dr Trefor please make a playlist about Partial Derivatives Equation, i think a lot of people will need it. thanks a lot

@DrTrefor 3 жыл бұрын

I plan to!

@ahmedmohamedkhogle6100 4 жыл бұрын

Nice keep going we need such those videos 🙂

@discoveryofphysics9303 2 жыл бұрын

Thank you Sir. It helped a lot.

@giuseppecammarata7528 Жыл бұрын

Thank you! Would it have been an error if I had written the symbol of the partial derivative in lieu of dx/dt and dy/dt (second row in final conclusion)?

@sandunmunasinghe9469 2 жыл бұрын

you are the best and i really like your videos

@continnum_radhe-radhe 2 жыл бұрын

Thank you sir 🔥🔥🔥

@hakaneskici2771 Жыл бұрын

Thank you for the explanation sir

@bilalabbad7954 2 жыл бұрын

Thanks

@abdullahmohamed2083 3 жыл бұрын

Thank you thank you sooo informative

@Avinash_Avee 2 жыл бұрын

Tnqsm 🙏

@vincentvaldinata2020 2 жыл бұрын

OMGG THANK YOU SO MUCH SIRR

@ossahmadrezaazimikohnabi5108 16 күн бұрын

This wss the highlight of my day. 🎉❤😭😍

@edwarddi3833 4 жыл бұрын

nice video, very informative! Thanks! however, what about the second derivative of f(x(t), y(t))?

@fullfungo Жыл бұрын

Just take the derivative twice

@joaomattos9271 Жыл бұрын

Amazing!!

@mathadventuress 3 жыл бұрын

hi dr trefor, can you please make a video or have you already on 2nd derivatives? say if z=f(x,y) and x=x(r,s) and y=y(r,s)...and we want to find the 2nd derivative of z with respect to x or y...i have an example in my book and i just dont understand it. i follow along the first derivative just fine. but then it says d/dr times the first derivative...which so far makes sense, but then it looks like it is doing implicit differentiation or something? please, thanks!

@akshitk6818 2 жыл бұрын

Really Awesome series of Calculus 3 and 4! Covers all the aspects that are usually taught in year 2! Your explanations are way better than my professor! Your videos have truly helping me in understanding better!

@DrTrefor 2 жыл бұрын

Glad you like them!

@ugursoydan8187 2 жыл бұрын

why we added the change due to y to change due to x? is there a proof of that?

@omer7895 2 жыл бұрын

What would be the gradient of w?

@realchatlabs 10 ай бұрын

Do I understand it right that the multi-variable chain rule the same as the single-variable chain rule (f . g)'(t) = f'(g(t))g'(t) with the difference only that you have a dot product between f'(g(x)) and g'(x) instead of a regular multiplication? (assuming that g(t) = [x(t), y(t), ...] to fit your examples).

@yoavmor9002 9 ай бұрын

Index notation and Einstein convention make this very easy to remember. But I don't think most folks are ready to display all variables as x⁰, x¹, x², x³, etc...

@frommarkham424 2 ай бұрын

it's useful in machine learning

@suvanshmahajan5902 3 жыл бұрын

Can anyone tell me if this is calculus of two independent variable(x,y) and here both these variables are dependent on t, so basically both are not independent of each other.How???

@DrTrefor 3 жыл бұрын

Yes in this situation we still call x and y independent even if they are both functions of the same variable.

@Nik-qh7cq 3 жыл бұрын

What if we have f(x,y) and a g(x,y) = (n(x), m(y)) and we need the derivative in respect of x for f(g(x,y))

@aditube8781 2 жыл бұрын

AMAZING

@doxo9597 3 жыл бұрын

I still don’t understand why it’s the sum of the two?

@sharifullahdanish4480 2 жыл бұрын

As you said we can directly solve it via putting the values of the x & y, so if you would do it, we will appreciate it. thanks..

@ronycb7168 Жыл бұрын

Thanks ... Also the chain rule looks a lot like the total differential divided upon dt

@HasanRaza-pe2sy Жыл бұрын

Great content. In india many of us study this in grade 11 and 12. As how i know as i am in 12 currently

@arpityadav.832 11 ай бұрын

burh it is of multivariable its teaches in college 1st . from which board r u?

@keerthanadiga3781 3 жыл бұрын

why single independent variable chain rule has product of 2 slopes why they had to multiply the 2 slopes?? can anyone answer .... it would be a chance for others to think

@enzoys Жыл бұрын

I mean, it looks like you can just substitute the t expressions in both x and y and have a normal chain rule

@user-jq4in7dm1v 2 ай бұрын

Still don't understand, why can't I just use dw/dt instead of partial

@khushinegi1363 3 жыл бұрын

What about the higher order derivatives in two variables?

@khushinegi1363 3 жыл бұрын

How will we calculate them?

@DrTrefor 3 жыл бұрын

Well we mostly do what we just did, but twice. So if you can do, say, the partial with respect to x then just do that again to get the second partial with respect to x.

@felixgomez5085 2 жыл бұрын

I have to tell you sir. You design usefull videos, with nice visuals. But you need to upgrade your sound device... It's mandatory at this level !!!!!!

@DrTrefor 2 жыл бұрын

i actually since have!

@slowsatsuma3214 4 жыл бұрын

I love you

@abnereliberganzahernandez6337 11 ай бұрын

I joined

@DrTrefor 10 ай бұрын

Hey thank you so much! Glad to have you:)

@Darkmatterkun 2 жыл бұрын

why does the theorem still hold true when the cross example proves it wrong

@josephhajj1570 4 жыл бұрын

Mr what's the proof of this chain rule😘😘😘

@altuber99_athlete 3 жыл бұрын

Great explanation! The pauses were on time, but maybe talking a bit slower can help the viewer to grasp the concept under discussion. Regardless, nice job! One question. In your example, we had "w = f(x,y)" (and particularly "w = x^2 y"), where at the same time "x = g(t)" (and particularly "x = 2 t + 1") and "y = h(t)" (and particularly "y = t^3"). That's fine, but I have a slightly different example. Suppose we have a function "w = f(x,y)" (without knowing any particular expression for "w"), where at the same time *"x = g(u,y,t)"* (not any specific expression for "x") and "y = h(t)" (not any specific expression for "y"). *Would the derivative "dw/dt" still be given by "∂w/∂x · dx/dt + ∂w/∂y · dt/dt, or would the formula change?* Notice the main difference between your example and mine is that in yours, "x = g(t)", while in my example, x = g(u,y,t)" where "u = i(t)".