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Find dz by implicit differentiation.(4x + 3y)1/3 2c...

Question

Find dz by implicit differentiation.(4x + 3y)1/3 2c

Find dz by implicit differentiation. (4x + 3y)1/3 2c



Answers

Find $d y / d x$ by implicit differentiation.
$x^{2}+x y-y^{2}=4$

To find d y over DX or the Druid of why start by differentiating both sides of the equation in respect to act. Okay, let's start with left side. The derivative of X squared is going to be two x and incidents in respect the exit find even leave like this. And now we have a product rule with X times. Why So when you do the derivative of X, which is one times why which is just why plus x times, the derivative of why which is one. But since we're driving in respect to X, we need to add multiply it by why prime with a derivative of why So we get that minus Ah, the root of y squared to why and again times wide prime because we're driving in respect to X. Okay. And now for the right side of the equation, uh, the the Trudeau of four is zero. Okay, now, in the next step, I'm going toe distribute all of my stuff out of the parentheses. Okay? From here. Now what I want to do is try to isolate these Ah, why prime? So I can solve for y prime for the Director of Why? So what we're gonna do is we're going to subtract everything from decided equation that is not multiplied by a white prime. Didn't get on the other side of the equation, which is two X And why then we're also going to factor out the UAE prime out of everything that's still here, which is this and this. Okay, who done here? So if we factor why prime out of ex my prime just X minus two. Why is equal to negative two x minus? Why? Because that's if we subtract two x and y for both sides of the equation, we'll get. And now in a divide both sides by X minus two. Why? To isolate Why prime? And you should end up with this. This is your answer of the Trudeau of why or why Prime Negative two X minus Y all over X minus two Y

If this problem were given in question. Expert always hurt before. And whereas find secondary, why must first find first Barrett off. Why? So let's take a look of all the traffic expert. Have to ex plus eight. Why and why? Primacy From this we see that weapon is going native x or y l Let's use kosher cruel and take their toe that one weapon will then be negative. Oh, or war waas. So minus X times or Y front divided by 16 wide screen here we know what y prime iss So we're gonna take this and implicated here that would give us negative off or wine or ex times actual. Or why divided by sting y squared basis Negative off or Wes Craig who? Less expert right by 16 for Cube. Now look at what we have here in the numerator and look at the orginal equation and those are the same. So we know that this is equal before, so we can then right y double time has negative four or 16 Like you. Yes, Well, guess allowed for the answer is negative warrant or or why you

Suppose you differentiate the equation for X cubed plus Ln of y squared plus two Y equals two X. With respect to X. Now using implicit differentiation we have derivative of for X cube plus a derivative of Ln a voice squared Plus the derivative of two Y. This is equal to the derivative of two. X knows that. Elena voice square. We can write this as two Ln of Y. and so for the first term we have the derivative 12 x squared plus. For the derivative of Ln of Y squared. Or to Ellen of why we have two times 1 over y times dy dx plus the derivative of two Y, which is two dy dx. This is equal to the derivative of 2X. Which is to now solving for dy dx we have To over y plus two times dy over dx. This is equal to 2 -12 x squared. And then combining two over Y plus two. We have two plus two Y over Y times dy over dx. This is equal to two minus 12 X squared. Now multiplying both sides by the reciprocal of two plus two, Y over Y. Which is Why over two plus 2 y. We have dy over dx which is equal to 2 -12 x squared. This multiplied by why over two plus two Y. Now simplifying that, we get dy over dx which is equal to two Y minus 12 X squared Y. All over two plus two. Y. We can further reduce this by factoring two from the numerator and nominator. We get Dy over DX which is equal to two times Y -6 X squared white, All over two times 1 plus Y. In which we can cancel out the two. And yet the reduced form dy over dx which is equal to y minus six X squared Y. All over one plus Y. And so this is the derivative of the equation

This problem, We're gonna be doing something What's known as implicit differentiation, which is very useful if we have multiple variables like y and X and a function. And we need to take the derivative of why with respect to X So we're given X squared minus four x y plus y squared equals four And what we do is we take the derivative of both sides. Now it's obvious that if you take the derivative of this site since the constant, it'll be zero eso. Now we want to focus on the derivative of this side right here. DDX So we know that it will be the same thing is taking the derivative this minus the derivative this plus the derivative of this. So what will end up getting as a result? Um, it's fairly clear here that what we'll get is, uh, two X, that's the derivative. Then we'll end up getting minus for why, But then, since we have, um, that right there, we'll have minus four. Why? And then Plus, uh, since we have the wire right here, what will end up having as a result is going to be a why prime, So multiply this by white crime. Then in addition, will have plus y squared. The driven will be two y I'm white prime So now what we have is two X minus four y minus for X Why prime? Because we want Thio ultimately have the change well done. So it will be minus four x times y prime plus two y times. Why prime? And like we said, that will be equal to zero. Then we can factor out the white prime here. So we do that. We end up getting a two x minus four y plus a negative for X plus two y with the UAE prime factored out equal to zero. Then we move over the two x and four. Why? So when we do that, we'll end up getting right here. But we have to switch the signs. So the plus for why minus two X Then we want to divide everything by this right here. So I'll take that great and divide it. We know that the queues can cancel out, so it will be a well cancel it out with a negative too. So this will be X minus two by over Xu acts minus wine. That will be our final answer for why Prime


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