5

Question 2Make sure to show all work in steps that can follow You do not need to algebraically expand or simplify derivative answers, but, If you do_ must be able t...

Question

Question 2Make sure to show all work in steps that can follow You do not need to algebraically expand or simplify derivative answers, but, If you do_ must be able t0 follow the work_In implicit or logarithmic differentiation problems you will need to solve for dyUse implicit differentiation t0 findfor the following curve72=(0- 5u)

Question 2 Make sure to show all work in steps that can follow You do not need to algebraically expand or simplify derivative answers, but, If you do_ must be able t0 follow the work_ In implicit or logarithmic differentiation problems you will need to solve for dy Use implicit differentiation t0 find for the following curve 72=(0- 5u)



Answers

$1-4$
(a) Find $y^{\prime}$ by implicit differentiation.
(b) Solve the equation explicitly for y and differentiate to get $y^{\prime}$
in terms of $x .$
(c) Check that your solutions to parts (a) and (b) are consistent
by substituting the expression for y into your solution for
part (a).
$$x y+2 x+3 x^{2}=4$$

We want to find white crime from this function. Solar's differentiate with respect two. X on both sides. We get four X plus one plants using for that rule for distance. So keep my eggs. Differentiate my wife. Let's keep my wife differentiate my ex. Just be one So it goes to zero. So I have white crime is minus for X plus one plus y over X Now for pet be. I want to find my wife straight from the function here, so just bring over whatever there is not off wine. So I have one minus two x squared minus X and this X I can bring over. So it's just over. X can erase this off you. So that's my wife. Okay, so for differentiating stripped from this one here when a different shape. Now, before differentiate, let me do it. Term by term. Explore minus one minus two X minus one. So why differentiate? I have minus X minus two minus two. So that would be minus one over X squared minus two. I can common base so our half minus one minus two X square factor writes off my minus sign. I'll just get this Okay, so for Patsy picking from here. So why Crime is minus four X plus one plus. Now for this. Why? I'm just gonna put this one in the all right or easier for me is to put this one in there, which is one of the X minus two X minus one over the entire X. When you simplify it, you can see that it will match what we have in above. So let's simplify it. And this is what I'll get. Four x and two X. Here. I will just get minus. Let me put a one of wax in front. So I have two x plus one over X. Bring my ex in. I have minus two plus one over X square. Now, this is the Samos. This this guy over here, which is also minus two. I can put my mind's outside plus one over X. So they are the same. So we have very fight that square story. So we're very fact that

We want to find white crime from this function. Let's differentiate with respect two x on both sites. So we're four x plus one plus using product will freeze the X differentiate the y now free. So why differentiate the X and one different ship with zero? So we have after simplifying we have This is our way. Crime. Well, part B, we want to find our wine street from the function. So it's just, um your algebra with the X Y across the one minus two X squared minus x. Bring this X over so it's over excellently erased off this accent key. Now we want toe find white Prime directly from this one here. So let me simplify these two extra power minus one minus two s minus one to simplify her. But why? Point will be minus. Expo minus two minus two. Rewriting and picking out the minus sign. I have this for Patsy. Okay, um, taking from here. Yah, taking from what we found directly in a So why prime? It's minus four x last one last. Now this white here, I'm going to replace it with the why I found in here. I'm just gonna put in one over X minus two X minus one Over here and open my exhale. Now simplifying you will realize that you will get this, which is exactly the same as what we found in B So our answer is verified.

Hi. Hey. We want to find white time from dysfunction. So let's different ship with respect to X on both sides. So we have nine two X minus two y don't forget toe differentiate the wild. The white prime equals to one when you defend should just be zero. So I will have over here. I have 80 next minus two wine. White crime equals to zero. So minus tool. Why? Why? Plan will be minus 18. So why Prom will be nine x over. Why? So let me call this the first equation for part B. We want to find why from the function itself. So just white Square will be nine X squared minus one. So why will just be plus minus square off? Nine X square minus one Now, differentiating from this directly my wife prime Now this really is power health. So differentiating directly, I'll just bring on the power and the power will. Whatever power is my response would be minus half differentiating inside the bracket. I'll have 80 next, so simplifying I'll get plus minus a class minus ni exe over square Ruoff nine X square minus one. Here for Patsy from the first part here. I just want to verifier my wife. Prime is nigh X over Y I'm just gonna put this. Why? Inside here. So here goes. I have nine eggs over plus minus square or nine X squared minus one, which is the same as over here, so we're verify our answer.

Yeah. So first we find the riveting here. We have eight X. Yes 18 Y Times Y. Prime Is equal to zero. So we try to get right prime by itself. So 18 Y. My time is equal to negative eight X. This means that by prime is equal to negative eight X. Cover 18. Why? This is a situation. Okay, so We can simplify this by divide by two. So we have 94 x over 91. So this is the relative. Mhm. So paul beat, we want to write why in terms of X. So the equation we have here, we move everything Mhm to the right to get right myself the same. So we have negative for X squared minus 36 plus two. Musics and then we divide by nine. Yeah. And then the square root. So what I say called uh square root native for Have squared last 36 overnight. Yeah. Okay. Okay, so we can keep yes, so we could simplify it but we will keep this for now. Mhm. And now we want to take the derivative of this. So let's see why prime is equal to So we have one half times for a square 36 overnight. This becomes sharp number needed 1/2. And you're multiplying by -8. Thanks. Okay, It's a very negative eight. Yeah, the nine is also yeah, 98 of the nine X. Okay. What? So we simplify this a little bit further. Let's see. We would get So this becomes eight becomes 94. So you have negative four. The top and X. And then The nine is outside the denominator. They have square roots. Yeah nine they had square root. Yeah. of maybe four x squared last 36 overnight. Okay. Yeah. So we can simplify this further and then nominally four X. And the nine here. But when we take it out it comes three so we take the square of that becomes three. So we have nine divided by three, that's three. And then we have four x squared plus 36. Should have simple find it. But it's okay for not. So this is the derivative For the 2nd part. Now we want to show that both of these derivatives are the same. So the way we found were going to substitute it in. So we the white prime from a While prime again is equal to 94. Yeah. Yeah. Over 91. Oh, who? Yeah. Uh We haven't 84 X. We know what why is nine times? Mhm Yeah, square roots. Mhm. Mhm of negatives. Yeah. For X squared plus 36. Have a life. Uh huh. Do you know what does this give us? Yeah, 94 X. Over. So we can do the same to the nine here. So we have three that we do make for X squared mastered six. And this is exactly like the derivative from harvey this the same as this derivative. So real good. Yeah. Yeah


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