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Question 11Find dyldx by implicit differentiation: x+2xlv?+v2 = 200B I U A - A - T = =...

Question

Question 11Find dyldx by implicit differentiation: x+2xlv?+v2 = 200B I U A - A - T = =

Question 11 Find dyldx by implicit differentiation: x+2xlv?+v2 = 200 B I U A - A - T = =



Answers

Find $d y / d x$ by implicit differentiation. $$e^{x y}+x^{2}-y^{2}=10$$

So when you're asked to find the derivative of something implicitly, it just means that it it's not a function, um, difficult. And, uh, yeah, what you need to do is differentiate in terms of your independent variable eso If you look at e to the X y, the derivative of that is still e to the X y, but then you need to multiply by the derivative of that, the expo. I guess I should say which is a derivative of this is a product role. I guess the derivative of X is one. You don't have to write it, why and then plus leave X alone, the drift of of why you have to say D Y d X right, because we're taking everything with respect to X. So it's almost like the chain rules. Another way of thinking that that the derivative of X squared is two x, The derivative of Y squared is to why, but again, with the chain rule, you have to say times the derivative of Y is equal to zero because adrift of of a constant zero. So what I would do to simplify this is go ahead and distribute that thing in, um, whoops. X E to the X y. I like to write d Y d excess after the fact. That's two X. I don't want to do too many steps at once and confuse you. So now what I would do is put the de y dx is on the left side beautiful. Okay. And subtract the pieces that did not have de y dx over to the right side. So now what you're doing is two steps you need to factor out the D Y DX and everything else stays the same. Mhm. I'll go ahead and rewrite this for you and then to solve. It's almost like thinking as a multiplication problem. Um, that you can divide it over, I think. And this is your final answer. Your answer key might be run slightly different. Um, but it's more or less this. Yeah, As I'm looking at the answer key all they did I'm just gonna change colors for you is a factor out, you know, divide everything by a negative one, and this is equivalent. It is acceptable as long as you multiply the top and bottom by the same thing. Uh, then you could do it that way. So this is ah, final answer.

We have X squared plus y squared equals 100. Now we're going to go ahead and differentiate with respect to X implicitly. So on the left hand side we have two X plus two y times the derivative of why, by chain rule equals zero because the derivative a constant zero. Now we can go ahead and subtract off the two x and divide by two y. That tells us that why prime is equal to negative X over why, and that's it.

Supposed to differentiate the equation, X squared minus three. Elena Boy plus Y squared equals 10 with respect to X. Now, by implicit differentiation we have the derivative of x squared minus the derivative of three Ln of Y plus the derivative of y squared This equal to the derivative of 10. From here we have two X three times the derivative of Ln of Y, which is one over Y times the derivative of Y, which is dy over dx. Plus we have derivative of Y squared which is to y times the derivative of Y, which is dy over dx. This equal to the derivative of 10, which is zero solving for dy over dx. We have -3 over why Plus two Y. This multiplied by dy over dx and then transferring to X to the right side. We get This equal to -2 x. And then from here we want to divide both sides by the coefficient of dy over dx. That means to divide this by negative three over Y plus two wine. Same here divided by negative three over Y plus two Y. And so we get dy over dx that's equal to -2 X over negative three over Y plus two Y. Simplifying this further, we get negative two X. All over negative three plus two Y squared over white. Which is also equal to negative two X times a reciprocal. Which is why? All over negative three plus two Y squared. And so we get -2 X. Y. All over two, Y squared minus three, multiplying the denominator by this negative, we then get dy over dx which is equal to two X. Y. All over three minus two Y squared. And so this is a derivative of the equation.

Dy dx for this. Why that's inside this function here or this equation. And we can't solve it for why? So we have to use implicit differentiation, which means taking the derivative without solving for why? All right. So the derivative of X squared is two X -3 times the derivative of the natural log of. Why? It's one over y times the derivative of Y plus the derivative of Y squared. It's two Y times the derivative of Y. And that's equal to the drift of of 10, which is zero. Okay, so everything that doesn't have a dy dx has to go to the right hand side. Everything that does have one can stay on this side. So I have negative three over Y dy dx Plus two. Y Dy DX Equals 2 -2 x. Alright. Now factor the dy dx out of everything over here. So minus three over Y plus two Y. And now divide out the stuff in the parentheses. Okay, You cannot leave uh three or four story fraction. So what I'm gonna do is make everybody look like a fraction by putting them over one. So the least common denominator is why? So I'm going to multiply everything by why? So with this one it cancels. So I get negative two X. Y over negative three plus two Y squared. And that is the derivative


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