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Find fy for f(x,y) = 4e% tan(6y)_...

Question

Find fy for f(x,y) = 4e% tan(6y)_

Find fy for f(x,y) = 4e% tan(6y)_



Answers

Find the four second partial derivatives. Observe that the second mixed partials are equal. $$ z=y^{3}-4 x y^{2}-1 $$

For this problem. We are given the function F of X. Y. Z equals coast of four. X plus three Y plus two. Zed. And we're asked to find F. Of or F. X. Y. Z. And F. Y. Zed zed. So to begin, we'll take the first partial derivative with respect to X. We'll need to apply the chain rule here. But we actually what I'll do is first take the derivative of the inside. So that's going to be for and then the derivative of coast is going to be a negative sign. So that's going to become negative for signs of forex plus three Y plus to shed. Then F. X. Y. Is going to be Now we multiply by three because that's the derivative of the inside with respect to Y. That's going to be negative 12 sign or not sign anymore. Now it's going to be coast negative 12 coasts of four X plus three Y plus two Zed. Then we'll take the derivative of that with respect to Z. So I'll first note that we're going to flip the sign again because we're taking the derivative with respect to close again. And then we're also multiplying by two. So we're going to get 24 sign of four X plus three Y plus two. Zed. Then we want that F wise, etc. So we start off by taking the first derivative with respect to Y. Which is going to give us negative three sign of four X plus three Y plus two. Zed. Then taking the second or taking the derivative of that with respect to said we're going to multiply the outside by two. So we get negative six coasts of four X plus three Y plus two Zed. And then we take the derivative of that with respect to Z again. So we'll multiply by two again and that's going to become positive 12 times sine of four X plus three Y plus two said.

Partial derivatives are extremely helpful because when we're in calculus and we want to optimize their minimize, we use derivatives. And oftentimes what we have is functions of multiple variables. Eso Therefore, we want to have partial derivatives to take the derivative of the different variables. So that's what's going to be the case for this problem here and were given a function. And we want to take two different partial derivatives that were given. The function that we have is the function f of X y Z, and it's defined by the co sign of four X plus three y class Cuzzi. And what we want to find is first, we want to find f X y z, so to find f x y z. What we're going to dio is first the partial derivative with respect to X um, and what we end up getting is a negative for sign four X plus three wide plus to Z. That makes sense so far. So then when we take F x y, then we end up getting a negative 12 co sign of four x plus three y plus two Z, and then lastly, we want to take the partial derivative of that with respect to Z, so we'll end up getting as a result is 24 sign, um, 24th sign time or 24th sign of four X plots? Three. Why us to Z? And that's what we'll end up getting. Then we want to calculate, uh, the partial derivative of y ZZ. So if we look at, um, the partial derivative of F let's respect why what we end up getting is a negative three sign, and we could just copy and paste this because we're gonna keep reusing it. So that's right here, then, um, f with the partial derivative Y Z will get a negative six code sign. Um, with this whole thing and then lastly, f partial derivative wise easy is going to be equal to taking that derivative of busy Variable will get a too negative, too, because it's co sign. So it'll go to negative sign. So the negatives cancel out and we'll be left with 12 sign of this portion again. And those would be our partial derivatives. I'm right here and also right here

In this problem, we want to find the second partial revenues for the function FXY is equal to the X. Tangent one. This question is challenging understanding of differentiation of multi varied functions. To solve what we must do is note that you find prosecutors, we use single variable differentiation techniques with respect to our function variable treating other variables is constant along the way. The example here showcases perfectly for the given function X. Y. And it's three perspectives. Thus, we use this principle to solve, looking for the second person F X X, F X, Y, F Y X and F Y Y. So let's first find F Xx and Xy to do so we have to first find fx so F X E X T N Y. Because the derivative E excessively the X. Thus fxx again GTX 10 Y and F X Y is by the tan derivative E V X. Against where Y. Next we find if Y X and Y Y. First we obtain fy as dxc can't square Y, which is the chance of derivative right away. F Y x is because you don't have any access itself can square Y. F y Y is by the chain rule. GTX to seek NYC. NY 10 YR two E X squared Y 10 Y.

For this problem, we are asked to sketch the graph of the surface given by F of X. Y equals four. So you can set up roughly our axes here. Let's say that's positive. Why that's positive access. Positive said, Well, if F of X Y equals for then what that tells us is that the zed value would just equal for no matter the value of acts or Y. So what this would look like be just a plain intersecting the easy to access at a value of four. Uh huh.


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