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$$\text { If } f(x, y, z)=3 e^{2 x^{2} y^{3} z^{4}}, \text { find } f_{x}(-1,2,1), f_{y}(-1,2,1), \text { and } f_{z}(-1,2,1)$$

$-96 e^{16} ; 72 e^{16} ; 192 e^{16}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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01:54

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01:12

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01:51

Let $f(x, y, z)=x^{2} y^{4…

02:17

So this one is kind of a continuation of number 20 because in number 20 we actually find all of these partial derivatives here. Um, so if you haven't done 20 or if you haven't watched the video on 20 I would go do that first because we can't necessarily just write this down. But once we get these partial derivatives, what we would need to do is just plug in like X is equal to negative one. Why is eager to two and Z is equal to one, and then that will give us the partial at those points. So let's go ahead and do this over here. Um, so it would be 12 times negative one and then two cubed is eight and then one to the fourth, just one and then e to the two times one because it's one negative one square and then 81 That should give us negative 96 e to the 16. And so then this is going to be our partial derivative with respect tax at that point. Now, over here, if we want to find the partial off this at the point again, we just plug in what we have or the point. So it be 18 the negative one squared, which is 12 squared four and then Z to the fourth, which is just one. And then again, we should just have to eat to the 16. Since the power there doesn't change it all on, then if we multiply all of that together, that should be 72. So seven to eat 16 So that this is our partial at that point. With respect to why and then for ze same thing. Just go ahead and plug in negative. 12 and 1 24 thieves negative one square one Too cute eight Z Q which is one and then again E to the 16 eso that is 1 92 and then e to the 16. And so this is our partial with respect to Z.

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