5

6 Fnd the_Integral 64-x dy...

Question

6 Fnd the_Integral 64-x dy

6 Fnd the_Integral 64-x dy



Answers

Use Equation 6 to find $ dy/dx $.
$ y \cos x = x^2 + y^2 $

Equation six says that if we're starting with the equation of the form capital F equals zero that D Y d x will be the negative partial derivative with respect tax over the derivative with respect away. So here we should rewrite this by setting it zero. So let's move everything toe left side and we see that equal zero. So we have this equation that we want and now we should go ahead and find this indicated derivative. So first will just go out and compute this numerator here. So we take the derivative of our new F with respect, Tex. So the first term that will give us e y co sign eggs, then minus one and then minus y oops. Now we go on to the denominator. So, hee, why derivative is just itself. You I still have the constant sine x there. There's no Y in the second term, so that derivative is zero. And in the last term, we have minus X. So therefore, using this equation six Keep my DX away first. Don't forget your negative out here so I'll just go ahead and multiply this by a negative one and the denominator here I'Ll just leave that as it is and there's our answer

Equation six says that to you my DX can be written as the following. Now, this is assuming that the original equation is of the form capital F equals zero. So in our case, we can go ahead and combine everything to the left hand side. So then we could just go ahead and call this our capital f Now we'LL use thie equation over here. This is Equation six from the textbook. So first, let's go ahead and find this numerator here. The rivet, um of our new expression capital if with respect to X So here we have X So first differentiate co sign and then using the chain rule. Well, excuse me. And for that my my ex in here and then using the chain room well supplied by the inside with respect to X and that'LL give us why and then both of these terms the negative one and minus sign Why those air constant with respect to X So the derivatives of zero Here's another derivative So co sign you get a negative sign this time will supply by the derivative of X y What respect? So why giving you x and then the ones of constant that goes to zero and then negative sign Why becomes negative coastline way? So let's put all this together And here you can also go ahead and just cancel out that double negative on the top. So if you do that actually, let's go ahead and cancel these negatives here we'LL get negative. Why sine x y And then in the bottom those both turn into a plus Signs. Yeah. No. And this right here. This is our final answer.

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.

We're being asked to simplify the given expression. To do this, we're going to use to distributive property, so we're gonna multiply both terms inside our parentheses by negative six x y Well, negative six x y times four x is negative. 24 X squared y Then we'll multiplying. Negative six x y bi Why? Which will give us negative six x y square. And because these air not like terms, we can't simplify any fervor. So our final answer is negative. 24 X squared y minus six x y squared.


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