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Determine the slope of the tangent line to the surface defined by $f(x, y)=2 y^{2}\left(13 x^{2}-3 y^{2}\right)^{5}$ at the point (1,2) formed when cut by the plane (a) $y=2,$ (b) $x=1$

(a) 1040(b) -472

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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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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So if we're trying to find the slope of this tangent, um, given we're cutting through these planes, why is he going to and access even though one what this is really telling us to do is to first? So since we're cutting through the plane, why is it would do Why is going to be a constant? So that means we're looking at the partial with respect to X at the 0.12 and then down here. Since we're cutting through the plain X as he could go one, this is going to be like we took the partial with respect. Why? And then we plugged in the 0.12 Um, but there's one thing we can do to make this a little bit even more simpler. Um, in each of these cases, we could just plug in. Why is he going to and then access to go toe one and then just take the I'll say the normal derivative kind of in quotes because it's still a partial. But so I'm just going to do the latter one, because having less variables floating around to me is normally easier, easier way to approach things. So let's go ahead and do that first, so I'm going to first find So why is always gonna be too So be x two eso We just plug in twos for all the wise eso to square times too. That's eight. And then we have 13, and then X is going to still be a variable. And then we have four times three, which is 12 to the fifth. So now we would just take the derivative of this with respect to X, essentially still the partial. So the f partial x to eso we would need to use chain rules. We do eight times five. So we get 40 13 x squared, minus 12. Raise the fourth and then remember, chain rule says, take the derivative of this S O. That should be 26 x on then. Negative 12 is just constant. And now, at this point, we could go ahead and plug in our other 0.1 or other number one, So be one too 40 times. So then one square times 13 minus 12 is just one to the fourth event. That will just be 26 um, and then 20 not 2040 times 26 is 1040. Yeah, And so this is going to be the slope of our tangent when we go through or when we cut through. Why is it going to at that point? Um and this is the same thing you would get. Like I said, if you were to just take the partial with respect to X first and then plugging the 0.12 Um, yes. So let's do the next one now. So over here. So now we want to first plug in X is a good one, And then why is going to be a variable? Um, so if we do that, that would only really change this to just make it 13. So does really supply it too much for us. But now let's go ahead and take the derivative with respect. Wire again. Essentially the partial. Um and so we would need to use chain rule here so I could just pull that to all the way out. And so we first take the derivative of why square to be two y times 13 minus three y squared race the fifth because we're having to do product rule. Ah, and then why squared states the same. And then we take the derivative of this, which again is going to be changeable. So it be or 13 minus three y squared. Raised to the fourth are not 45 out here. And then we take the derivative of the inside, which would be negative six y And then let me just clean this up a little bit. So I'll just keep the two on the outside there. Thin to why? 13 minus three y squared raised the fifth. And then plus, we're actually not plus minus. It would be negative. 30. Um, Why cubed 13 minus three y squared, raised to the fourth. All right. And now we can go ahead and plug in to so yeah, f sub y of 12 eso if we plug into that would give us so four. Um, so again, that would just be one to the fifth. And then minus 30 times too cute or actually, I would just be eight. So All right, this is minus 2 40. And then this would just be one to the fourth. Uh huh. Um, So, actually, we just get rid of these ones here, and then that would be two times four, minus 36 or four minus 2. 40 is 2 30 negative to 36. Multiply that by two. And that gives us negative for 72. And so then our soap of our line when they cut through excessive goto one at 12 is going to be negative. 472. And so again, like I said, you could first find these partials and then plug in either one or two to Pentagon, which you're keeping constant. Um, but I always just think it's easier to do it this way because I don't like to see a bunch of variables floating around, but no matter how you do it, you should still get these two answers here.

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