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6. [0/2 Points]DETAILSPREVIOUS ANSWERSSCALCCC4 11.5.030.Use the following equation (or any other method) to find €z/dx and dz/dy_ aF 2F dz dx dz dy dx @F dy @...

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6. [0/2 Points]DETAILSPREVIOUS ANSWERSSCALCCC4 11.5.030.Use the following equation (or any other method) to find €z/dx and dz/dy_ aF 2F dz dx dz dy dx @F dy @F dz 8zXyZ = tan(x + Y + 2)dz/xaz/8yNeed Help?Read ItTalk toa Tutor

6. [0/2 Points] DETAILS PREVIOUS ANSWERS SCALCCC4 11.5.030. Use the following equation (or any other method) to find €z/dx and dz/dy_ aF 2F dz dx dz dy dx @F dy @F dz 8z XyZ = tan(x + Y + 2) dz/x az/8y Need Help? Read It Talk toa Tutor



Answers

Use Equation 6 to find $ dy/dx $.
$ e^y \sin x = x + xy $

If you think the equation six, we first have to rewrite our equation in the following form. And then if we do this, then Equation six tells us a formula for the derivative we're trying to find. So let's go ahead and rewrite this so we could obtain F. So notice that if you just subtract, we get zero. So let's just go ahead and call this new equation this new function capital F Sorry about that. I ran out of some room near me back up a capital F. So looking at our equation over here, there's some derivatives we have to find. So let's go find those. So first is the derivative with respect to X so tan in verse and then multiply by the derivative of the inside with respect her ex using the chain rule there. And then we have minus one and also a minus to exploit now for the denominator so very similar for the first one so derivative of Arc Tan square that term in the bottom. And then we'll supply by this term on the inside the derivative of that with respectful why and then here derivative with respect a Y zero and this one negative x clear. So now let's put these together and also noticed the negative sign out here in the very front So we can just go ahead and take this negative sign and just cancel that. Oh, cancel that out. And then we'LL also end up including a negative over here. So we get negative to X y plus one plus two ex wide and let me include that denominator for the first term just for the first time there and then in the denominator. And if anything, you could probably just go ahead and get a common denominator and simplify. But otherwise, here's our final answer.

Let's use Equation six, which assumes that your equations of the form capital F equals zero. So in our case will first just rewrite this by putting everything on one side. So let's write. This is why co sign minus X squared minus y squared. And by doing this, we see that equal zero. So then equation six. Is that do I d x is given by the following Oh, so we'LL go ahead and compute these the riveters So this one first So we'll go out in different she our new function capital f with respect to X here. So you have co sign that'LL give you a negative sign and then he's the power rule here Now, on to the next one, this time with respect to why we just get the Constant co sign and then he's the power rule and putting these together And don't forget the negative sign over here. So I'll go ahead and just cancel this negative sign with these two negative signs over here. So finally, by Equation six and there's our answer, Yeah,

A problem. 10. So we don't want to find the wreckage of that. I just re copy the problem, So just take the directive. Fax. This equals toe. Uh, they're all the same. Fine sea and us. Why does one aware squirts minus X square one box? Why is zero and these pie or six? So that's, uh, one times because I power sex aids escorts three over two. And plus these deeds one. Right. So it's just one scores three over two or the why that this ico through there are the same. So that the same one course. Really where to plus are crossed. Science X right When actually zero to see zero. So? So this is zero. But the doctor. Why, it's just works way over. So and how about yeah, see, it's Nico school. Oh, that's a constant. No. On times minus okay. And for us Oh, here's here's Nazi. Here's a zero right minus. Sci fi's minus one, huh? And this is the one. So it's minus one, huh? So it's like a factor. So here's our answer

Whether in this problem, we're have to find the Grady int of this function f at the 0.0 pi over six. So let's remember the Grady int of efforts to find as from a vector whose components are f sub x f supply and F sub Z. So that means we're going to want to start by finding these partial derivatives. So let's start with F sub X treating. Expect the variable on the other two as Constance. Let's see what we get now is co signed Z. It's just going to be treated like a constant. The derivative of E to the X Plus y is itself, but then we need to multiply. But the general by the partial derivative of X plus y with respect to X, which is just one so that ends up not affecting things. Plus, here we have this y, plus one is just the constant that stays in our derivative and now the derivative of in verse, Sign of X, with respect to X, Let's recall is one over the square root of one minus X squared. So there their first component. Now, if some why once again, this coast signs e is a constant derivative of either the explosive wise itself. This time you mark multiplied by the partial of X plus y with respect to why which again is one and has for the second term Ah, this in verse sign of X is treated as a constant and the derivative of why plus one is just one so that won't affect anything. And finally, our third component of sub Z here this eat of the X Y is now treated as a constant and the derivative of Cose NZ is minus Sign C. And finally, there are no Z's in this second term, so that will give us partial route of zero, which is nice. Okay, so let's clean this up a little bit so we can plug in easier. We'll get E to the X plus why Times Co sign Z plus Well, I plus one over square root of one minus X squared. That's our first component. Our second component will be e to the X plus why? I'm just copying mainly and cleaning up a little bit as we go. But there's not too much to do here e to the experts. Why, at times co signs E plus in verse. Sign of X. You know, uh, getting nothing to do but copy Really? Here minus e to the experts y times sine z Okay, time to plug in. We want the Grady and specifically at the 0.0 However, six. Let's just plugging zero frack zero for Why Empire for six? Busy If we do that, if X and y above zero e to the zero is just one. So that will Let's give us one times. Coastline of Z co Sign of pi over six is squared of 3/2 came now plus why plus one y zero. So that's just +11 minus X squared. Here is one minus zero, which is One squirt of one is one. All right over here again. He zero is just one co sign of see again Squared of 3/2. Plus in versus sign of zero is zero. Since signing 00 Finally, over here we have minus again, he goes. Zero was one. In this time you have the sign of Z Sign of five or six is 1/2. Okay, so we have square 3/2 plus one. We can simplify that a little bit. Look at Skirt of three plus two over to. So that is our first component. Second component we can clearly see is just squirt of 3/2 and 1/3 component. Negative one times 1/2. Give us negative 1/2 and we are done. This right here is the great inspector at the point that we were asked for hope.


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