The Multi-Variable Chain Rule: Derivatives of Compositions
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- เผยแพร่เมื่อ 22 มิ.ย. 2024
- 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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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.
So glad it helped!
Why he isn't famous yet ?
Great explanation sir.
Really I just love your videos. 🤩
He is famous. You may need to remember that not many people like or do mathematics, at least on TH-cam, 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
Because he has a difficult accent which non native speaker can't understand
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Content and quality of video is masterclass, hope to see a rise in quantity of other concepts as well. Thank you
these videos are so good... i hope you are dreaming big enough with your channel. This is the future of math education
Brilliant! I can't believe how you made this concept extremely easy!
Great video as always. It helps me a lot to rebuild my math background. Keep it up!
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!
You are an excellent teacher!!! Keep up the good work.
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
Great videos man, you really have a very charismatic way of explaining math.
u are really doing great sir. giving the real insight to mathematics .
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
Beautifully explained sir.
Love you.❤️
Impressively simple yet effective expanation!
Great explanation!!
A very nice explanation of the total derivative.
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 .
Great exposition of a difficult concept, Dr. Bazett
Subs from India
This is how math should be taught.. Just superb.
Thank you!!
I loved the tree diagram. Great job sir 👍🏻
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.
Thank you Sir. It helped a lot.
Best maths teacher ever
Thank you for the explanation sir
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!
Nice keep going we need such those videos 🙂
Great content! Greetings from Poland!
Hello Dr Trefor please make a playlist about Partial Derivatives Equation, i think a lot of people will need it. thanks a lot
I plan to!
Thank you thank you sooo informative
you are the best and i really like your videos
Amazing!!
Thank you sir 🔥🔥🔥
OMGG THANK YOU SO MUCH SIRR
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!
Glad you like them!
Tnqsm 🙏
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)?
nice video, very informative! Thanks!
however, what about the second derivative of f(x(t), y(t))?
Just take the derivative twice
Thanks
it's useful in machine learning
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...
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!
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).
I still don’t understand why it’s the sum of the two?
AMAZING
Great content. In india many of us study this in grade 11 and 12. As how i know as i am in 12 currently
burh it is of multivariable its teaches in college 1st . from which board r u?
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..
Thanks ... Also the chain rule looks a lot like the total differential divided upon dt
why we added the change due to y to change due to x? is there a proof of that?
What would be the gradient of w?
I mean, it looks like you can just substitute the t expressions in both x and y and have a normal chain rule
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 !!!!!!
i actually since have!
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???
Yes in this situation we still call x and y independent even if they are both functions of the same variable.
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
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))
What about the higher order derivatives in two variables?
How will we calculate them?
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.
Still don't understand, why can't I just use dw/dt instead of partial
Mr what's the proof of this chain rule😘😘😘
I joined
Hey thank you so much! Glad to have you:)
why does the theorem still hold true when the cross example proves it wrong
I love you
You're a great teacher really, but your voice isn't clear and it sounds like a little noise there. Try a better mic. I WISH YOU THE BEST!
Honestly, I don't really care about how and why this works on a fundamental basis. I am just looking to apply it correctly..
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)".
hhhha
Good than common teacher ,bu also not clear and to some degree, is a waste of time and energy to watch
th-cam.com/video/0Ppc7AzJtvI/w-d-xo.html
full concept of chain rule