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

(a) $80 x y^{3}\left(2 x^{2}+3 y^{4}\right)^{4}$(b) $12 y^{2}\left(2 x^{2}+3 y^{4}\right)^{4}\left(2 x^{2}+23 y^{4}\right)$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Missouri State University

Campbell University

Harvey Mudd College

Idaho State University

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 in order for us to take the partial derivative of this s O for the first one, they want us to do the partial respect to X Adele by Dell X. Now we're going to assume these wise here are constants with respect to X. So any kind of property we would have just like a single variable case where we take the derivative of some constant will apply those here so first. Well, right. This is f sub X to kind of represent that this is the partial with respect, X. And now remember, we can factor out this for White Cube because it's a function multiplied by a constant. So before y que del by del y of two x squared plus three y to the fourth race If it and now over here to take this derivative, we could use changeable so before Why cute and then So we've used powerful. First we moved the five out front two X square plus three y to the fourth, um, raise to the fourth power because we subtract went off and then we take the derivative on the inside here. So remember this. Why here? We treat as a constant. So when I go to take the Dorado of the inside, it would be or what are for X And then why? To the fourth power times three. That would all be a constant. So it be plus zero. And now we can go ahead and just multiply everything together. So it would be 16 x y cute? Well, actually about 16, because I almost got that five. Eso would be 20 times four should be 80 x y keep and then two y square plus three wide the fourth all race for So this is going to be our partial with respect Thio X now to get our partial with respect to why will apply the same idea. But now, instead of treating the wisest Constance, we're going to treat the excesses, constants or Dell by Dell. Why? So this right here is our only constant in this case, at least for the variables. So now we have a function that depends on why multiplied by a function that depends on why. So we need to use product rule. So this is going to give us so we have f sub y are partial of act with respect to y and would use. Probably cool. So I'll just write that out first for White Cube. No, by the or Y of two x squared plus three y to the fourth to the fifth. And then plus, and then we just flip those two x squared. Plus two a freeze here, three y to the fourth raise the fifth in Del by del y of four y cute. So this first one again, we'll just use chain role in power rule. Still be five two x squared plus three y to the fourth now to the fourth power. And then we take the derivative on the inside. And again we treat this X here as if it's a constant. So the derivative of that is zero. And then the derivative of three y to the fourth is going to be plus 12 wide cute on, then over here. Actually, let me expand the screen a little bit. So we just right that down first and then four y que we just use power rules would be 12 y squared. Okay, so now let's go ahead and factor some stuff out, and actually, I just go ahead and erase this zero plus here and see what we can factor. So I can factor out a 12 from here and here. E could also factor out a y squared from here in a y squared from there. So that would just be one wide. Now, um, so the super to go ahead and factors that out, we have 12. Why squared and then also weaken Factor out four copies of these. So be two x squared plus three y to the fourth, raised to the fourth. And then everything we would have left. So remember, we still have one y left over from here so we can multiply that up front here. So would be 20. Why? To the force. Um, plus, and then we just have one copy of this over here. So would be to expert plus three y two or three that we could go ahead and combine the Y to the horse. So give us 12 y squared two X square plus three y to the fourth to the fourth and then 23 y to the fourth plus two x squared. And so then this here is going to be our partial with respect to Why eso again. Remember when we're taking these partials, just whichever partial you take, You treat all of the other variables just as if they were a constant.

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