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Determine the slope of the tangent line to the surface defined by $f(x, y)=\left(4 x^{2}+3 y^{3}\right)^{7}$ at the point (1,-1) formed when cut by the plane (a) $y=-1,$ (b) $x=1$

(a) $y=56 x-57$(b) $y=63 x-64$

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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Find the slope of the tang…

So if we want to determine the slope of the tangents that passed through these two planes here, essentially, what this is telling us to do is to really just fine. So this first cases were keeping Why a constant since why is always gonna be negative one. This is the partial with respect to X at the 0.1 negative one and then over here. Since we're keeping exa constant, we're taking the partial with respect to why so the f sub y of one negative one. And so something else we could do just to kind of make the calculations a little bit simpler. Um, we could actually plug in these numbers to start since remember, we're assuming why is a constant or X is a constant and then just take the derivatives with respect, just X or Y. And it's kind of like we don't even do the partial derivative, um, in some sense. So that's just what I'm going to do, because I don't like having to look at tons of variables if I don't have to. So, over here, the first thing I'm going to do is first just find what is f of X negative one. Because remember, all values of why are going to be negative one in this plane. So we come over here and just plug those in s. I would give us four x squared thing. That would be negative. One cube. So that just becomes negative. Three. And then this to seven. And now we could take the derivative of this book perspective X, essentially still the partial. But we'll just write that here. So taking the derivative, we would use chain rules. So it be it looks like seven four x squared, minus three, raised to the sixth. And then remember, chain rule says we take the derivative on the inside, which would be, um, eight X and then derivative of three is just zero. And now we can go ahead and plug in one, so f so x one negative one and we can multiply 78 and one to go. So that gives us 56. That would be 56. Oh, that's a three. Um, and then for minus three was once. That's just 1 to 6. So this ends up being 56 for the slope of our tangent line. Yeah. Now, over here. we could do the same thing. But what I'm going to do instead is plug in X is equal to one first. Because, remember, X is going to just be one in this entire plane. So this is going to be f of one. Why? Which would give us four plus not three or plus three y Cube race in seventh. And now we can go ahead and take the derivative. This with respect. Why? Essentially again? Still the partial. And then this would give us, um so seven times four plus three y cubed. And then we take the derivative on the inside. We should give us nine. Why squared now? We can go ahead and plug in negative one for why? And then that would give us oh seven and then four minus three again. So that's just one. This is toothy six power here and then over here, negative One square would just be nine. So then this is a being 63. So the partial, um, at that point, with respect to y 63 or essentially the slope of the Tangela is 63. So again, you could have actually just went ahead and first found the partial with respect to X or with respect, why and just went from there. But since we're only really interested in these two planes here, I can just go ahead and plug in the number, and it just makes things a little bit simpler. But if you take the partials and then plug in one and negative one for X and Y respectively, you'll end up with the exact same solution.

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