Multivariable Calculus Episode XXX: Return of the Grant Also, a chain rule intuition prediction: think of the parametric functions x(t) and y(t) instead as a vector valued function r(t) = x(t)i + y(t)j, so from the previous few videos, we know that a tiny nudge dt along the curve can be represented by the vector derivative r'(t) = x'(t)i + y'(t)j (which is tangential to the curve). Then think of f(x(t), y(t)) as projecting the curve onto some surface in 3d, so the total derivative df/dt can be thought of as a directional derivative: nabla of f dot dr/dt which is ∂f/∂x*dx/dt + ∂f/∂y*dy/dt
Thank you SO MUCH for this video! I am learning the Slutsky Equation for a microecon class and positively no one was explaining why the derivative of the compensated demand function played out like it did. The sum of all the terms of the multivariable function was totally glossed over. This video clearly explained how each term is treated in a multivariable function, and I am incredibly grateful, because this problem has been seriously bothering me.
Hi Grant! I never noticed you were teaching on KhanAcademy. I'm a big fan of 3blue1brown videos and started to unravel the secrets of physics as an amateur. Thank you so much for this video, I always loved your way of explaining and am very exited to see what you will come up next. This video was just so great!
@@3blue1brown wow I have been seeing all the comments about you since the start of the series and woah you replied here. I know i am a little late to discover you but I am here after your linear algebra series and it's wonderful how many people love you and yeah I LOVE YOU TOO MAN
this is equivalent to directional derivative of f(x, y) in the direction of the tangent vector of the curve traced by x and y, the formula in the green box is just ∇f ⋅ (dv/dt) where v = xi + yj
I wonder if ∂x in the denominator and dx in the numerator can cancel each other. I suspect maths purists will say it's incorrect to do so, but I feel it makes sense to do so, because it leads to a meaningful result that agrees with intuition.
It's like nesting into a list using indexes in python. First index into the primary function, then index in to the secondary function within the primary function.
if you view del y and dy in a similar way, you can see they cancel out. and that goes for x too. (a similar method is used in differentiating single var parametric functions) in the end ur left with two rates of change (signifying the two dimensions) and we sum them up to get the final result you probably wouldn't call this a rigorous proof but it does generalize it
There again, please don't split it. This playlist has a great flow, and I've been learning loads from it. That we hear two different presenters is just fine. Thanks to both of you.
![The Chain Rule for Derivatives in Calculus - [1-5]](http://i.ytimg.com/vi/HpymsAXhJAM/mqdefault.jpg)
![YoungBoy Never Broke Again - Missing Everything [Official Video]](http://i.ytimg.com/vi/dTYeMZGzOzw/mqdefault.jpg)
YES! 3blue 1 brown! Khan Academy really is getting the best of the best.
I literally gasped when I recognized your voice! Sort of like seeing a celebrity in your hometown.
Sir you are awesome.
Same
Multivariable Calculus Episode XXX: Return of the Grant
Also, a chain rule intuition prediction: think of the parametric functions x(t) and y(t) instead as a vector valued function r(t) = x(t)i + y(t)j, so from the previous few videos, we know that a tiny nudge dt along the curve can be represented by the vector derivative r'(t) = x'(t)i + y'(t)j (which is tangential to the curve).
Then think of f(x(t), y(t)) as projecting the curve onto some surface in 3d, so the total derivative df/dt can be thought of as a directional derivative: nabla of f dot dr/dt which is ∂f/∂x*dx/dt + ∂f/∂y*dy/dt
Thank you SO MUCH for this video! I am learning the Slutsky Equation for a microecon class and positively no one was explaining why the derivative of the compensated demand function played out like it did. The sum of all the terms of the multivariable function was totally glossed over. This video clearly explained how each term is treated in a multivariable function, and I am incredibly grateful, because this problem has been seriously bothering me.
Hi Grant! I never noticed you were teaching on KhanAcademy. I'm a big fan of 3blue1brown videos and started to unravel the secrets of physics as an amateur. Thank you so much for this video, I always loved your way of explaining and am very exited to see what you will come up next. This video was just so great!
+R3fini Thanks man! Do let me know once you've unraveled all the secrets in physics :)
@@3blue1brown wow I have been seeing all the comments about you since the start of the series and woah you replied here. I know i am a little late to discover you but I am here after your linear algebra series and it's wonderful how many people love you and yeah I LOVE YOU TOO MAN
Thanks a lot. I hate simply following the formulas I don't understand the meaning of. This video made it clear!
3Blue1Brown????
y e s
Grant is back!!
Jake Waitze yay!!! Exactly
this is equivalent to directional derivative of f(x, y) in the direction of the tangent vector of the curve traced by x and y, the formula in the green box is just ∇f ⋅ (dv/dt) where v = xi + yj
You are back finally. I've been watching.
Would have been nicer to use an example where this chain rule is not an 'obvious' consequence of the product rule. Something like log(x+y).
I agree
I really enjoy your videos! You have become my new favorite Math prof. =D
I know this guy is a very accomplished mathematician but I still just feel safer when I hear Sal is teaching the lesson haha
Yay! Grant is back.
I wonder if ∂x in the denominator and dx in the numerator can cancel each other. I suspect maths purists will say it's incorrect to do so, but I feel it makes sense to do so, because it leads to a meaningful result that agrees with intuition.
For a moment I was confused on how come 2*y is 2sin cos. Later I realized it to be small x :-). Neatly explained Thanks.
It's like nesting into a list using indexes in python. First index into the primary function, then index in to the secondary function within the primary function.
Best teacher ever - only this guy should do all the videos
Oh I just played this video only looking at its title, and I thought I was watching on 3blue1brown!
This was great. I think a good application video would be the derivation of the Navier Stokes equation (or at least a portion of it) using this rule .
After watching Sal writing in the last several videos, now I’m sure Grant is not writing with a mouse. 😆
You cannot just give an example and then generalize it into a rule. I know that the Multivariable chain rule is correct, but we need rigorous proof.
if you view del y and dy in a similar way, you can see they cancel out. and that goes for x too. (a similar method is used in differentiating single var parametric functions)
in the end ur left with two rates of change (signifying the two dimensions) and we sum them up to get the final result
you probably wouldn't call this a rigorous proof but it does generalize it
Marvellous💯
If both X and Y are dependent on X isn't f a simple single variable function
Yes, but this is particularly useful in differentials
So is the derivative of a multi variable function just the total differential divided by dt ?
Holy frog, it's Grant/3B1B
you searched for it, it was not recommended
If f = x^2 + y^2 .. what will b the derivative of main function...w.r.t x and y
grant is back😀
I never usually comment but thank you.
wait.. isnt this the guy from 3blue1brown?
yes
You made it easy
3Blue1Brown! No wonder his voice sounds familiar!
great!!!!
Hey Grant, What is the name of this software that is used for typing?
3blue1brown ... is it you ???
It's grant
Wow!!!
Yay!
yay grant!
3b1b god
which video is 3 blue brown and which is sal khan. please make two separate playlists.
There again, please don't split it. This playlist has a great flow, and I've been learning loads from it. That we hear two different presenters is just fine. Thanks to both of you.
Just look at the channel name for god sake