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$$\begin{aligned}&\text { If } f(x, y, z)=4 x^{2}+2 y^{3}+5 z^{5}+3 x-2 y+11 z+12, \text { find }\\&f_{x}(1,-2,-1), f_{y}(1,-2,-1), \text { and } f_{z}(1,-2,-1)\end{aligned}$$

$11 ; 22 ; 36$

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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05:22

For $f(x, y, z)=4 x^{3} y^…

02:33

$$\text { If } f(x, y, z)=…

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$$\begin{aligned}&…

01:51

Let $f(x, y, z)=x^{2} y^{4…

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find $f_{x}, f_{y},$ and $…

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06:53

Find $f_{x}, f_{y},$ and $…

So this is kind of a continuation of number 16 because we actually found the partial derivatives in that question. Um, so this is what we ended up finding from number 16. So if you haven't watched that video, I would go back and watch that one. Or at least attempt number 16. Um, and then see if you get these same partial derivatives. But with that in mind, all we need to do now is plug in these values. So this is like X, Y and Z is when we come over here to plug these and actually the me spent the screen a little bit, so f x one negative to negative one. So this is just like when we would plug in for finding the derivative at a point in the single variable case. But now we just plug in one number for each of these. So that would just be eight times one plus three. So that's just 11. Thank half of why one negative to negative one. Eso We only have a wise so we just plug in negative too. So that would be six times four minus two, which looks like could be 22 and then, lastly for F Z one negative to negative one. So we only have a Z here, so we just plug negative one. And so that would just be 25 plus 11 or 36. So our partials, all of the years points here would be 11, 22 36. So again, if you haven't done number 16, I would go watch that video first so you can see how we got those three partials because otherwise just kind of writing these down isn't really to sufficient, because you should show how you've got these first.

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