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## Multivariable calculus

### Course: Multivariable calculus>Unit 2

Lesson 5: Multivariable chain rule

# More formal treatment of multivariable chain rule

For those of you who want to see how the multivariable chain rule looks in the context of the limit definitions of various forms of the derivative. Created by Grant Sanderson.

## Want to join the conversation?

• Did Grant ever get around to making videos on the connection between Linear Algebra and MV Calc, like he mentioned at the end of the video? Or the more general version on the MV chain rule?
• I don't see how this is equal.
• I'm assuming you are asking about the connection Grant mentions with the formal definition of the directional derivative. The whole limit that Grant writes in Magenta is really the directional derivate, because if you go back to video he mentions, the vector "a" is "v(t)" in this case, and the nudge "hv" is "h*dv/dt" (remeber this is a vector from Sal's videos from the previous section), so really it's all just the same! And that's where the directional derivative pops out.
To be honest I wasn't seeing it at the beggining but I do see it know and it's kinda amazing :)

Cheers
• At , how did Grant arrive on this expression ( involving the error term ) from the previous expression involving limit ( The formal definition of derivative )?
• It would be interesting to look at the formal proof why at you can just cross o(h).
• what if f(v(t)) is itself a vector valued function? gradient of that is not defined.
• Assuming that you are saying the derivative of f(v(t)) with respect to t. If that is so, then you can write the function as a sum of a scalar valued functions that depends on r(t) multiplied by the corresponding basis vector. Since the derivative operator is linear, then it distributes over this sum, hence you would use the chain rule per component.
More tangibly, suppose that you defined this function as <p(v(t)),q(v(t))> in two dimension. Then surely the derivative will be d/dt(p(v(t))i+d/dt(q(v(t))j, and then we would need the chain rule at that point.
• At around minutes, we ignore `o(h)` because it's much smaller than `h`. I can understand `o(h)/h -> 0, as h-> 0`, but here `o(h)` is part of the input to a function `f`. It's not immediately clear how we can ignore this.
• When you are re-writing v(t + h) from the above expression, shouldn't it be (-)o(h) or maybe it just doesn't matter?
• Are there any videos on this with worked examples?
• At - formal definition of directional derivative part.... we should take unit vector along v' (v' cap) and not v' ... I guess there should be a correction