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Find the partial derivatives with respect to (a) $x$ and (b) $y$.$$f(x, y)=3 x^{4}-2 y^{3}-7 x^{2} y^{5}+8 x^{2}-3 y-5$$

$12 x^{3}-14 x y^{5}+16 x$(b) $-6 y^{2}-35 x^{2} y^{4}-3$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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Lectures

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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So if we want to take the partial this perspective of X and y, remember what we're going to do? So if we were to start off by doing the partial of this respect to x o Dell by Dell X, What we assume is all of these wise here are going to be constant. So just like how we would see is like five or three or eight as a constant. Every why we see here we're going to treat just like it was a constant. And we're just taking the derivative of this with respect to X like we did in single variable calculus. So now we can come over here and so on the left. Since we're doing the partial with respect to X weaken, right, this is F partial X, which we normally just right is the subscript X and then over here. So we just take the derivative just like we would normally for X. And remember, we just treat why, as some constant so three x to the fourth abuse powerful. So it be 12 x cute. Now, since why is it constantly Cuba constant and then multiplied by a number that's still a constant. So the derivative of to like with respect to X for the partial that is going to be zero. Now, over here. Just like how we would pull out this negative seven. We can pull off the why to the fifth and then just take through of Why Square? So the seven. Why? To the fifth. And then we take the derivative of X squared, which is going to give us two X and then plus so eight x squared. Just use powerful. So 16 x three wise. So a constant. Why times another constant is still going to be a constant zero. And then negative. I was just a constant. So plus zero, Then we could go ahead and clean this up a little bit. So are partial with respect to X is going to be 12 x cubed. And I'm just going to move the X out front of the why on then multiply. The two and seven will be negative 14 x y to the fifth plus 16 x. So this here is going to be our partial with respect to X Now, to get the partial with respect to why we're going to do the same thing. So let me just go ahead and stand a little bit. So we're going to do Del by Dell. Why? And now all of these X is we're going to treat as if those air constants. So we would right f partial with respect to why and this is going to be equal to. So if we have a constant raised to a power multiplied by a power will still be a constant. So this will be zero and then two week we just use powerful, so minus six y squared. Um, so again, seven times expert, that's a constant. So we pull that out and then just take the derivative of why to the fifth, which would be five y to the fourth. Then we take the derivative of so constant square times a constant so constant, so plus zero negative three y that just the swift negative three and then the derivative of negative I will just zero that we could go ahead to clean this up so or partial with respect to why is going to be so negative. Six y squared negative 35 x squared y to the fourth minus three. And so then this here is our partial with respect to why

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