Multivariable chain rule

This is the simplest case of taking the derivative of a composition involving multivariable functions.

[acruzp]

YES! 3blue 1 brown! Khan Academy really is getting the best of the best.

[johnkent]

I literally gasped when I recognized your voice! Sort of like seeing a celebrity in your hometown.
Sir you are awesome.

[ShirimeCid]

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!

[chiquita_dave]

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.

[harry_page]

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

[kassymchurchman]

Thanks a lot. I hate simply following the formulas I don't understand the meaning of. This video made it clear!

[peppercanfly]

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

[sols-ee]

I really enjoy your videos! You have become my new favorite Math prof. =D

[oneinabillion]

You are back finally. I've been watching.

[zakariaabdi]

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.

[dawne]

I know this guy is a very accomplished mathematician but I still just feel safer when I hear Sal is teaching the lesson haha

[shrinandhanbk]

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.

[dmt_benchpress]

Best teacher ever - only this guy should do all the videos

[짤막-xl]

Oh I just played this video only looking at its title, and I thought I was watching on 3blue1brown!

[joluju]

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.

[lazer]

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 .

[cauchyschwarz]

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).

[Anteater]

So is the derivative of a multi variable function just the total differential divided by dt ?

[happywater]

After watching Sal writing in the last several videos, now I’m sure Grant is not writing with a mouse. 😆

[afnankhan]

If f = x^2 + y^2 .. what will b the derivative of main function...w.r.t x and y