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1. Use implicit differentiation to find an equation of the tangent line to the curve at the given point:x2 + 2xy -y2 + x = 20 (4,8)...

Question

1. Use implicit differentiation to find an equation of the tangent line to the curve at the given point:x2 + 2xy -y2 + x = 20 (4,8)

1. Use implicit differentiation to find an equation of the tangent line to the curve at the given point: x2 + 2xy -y2 + x = 20 (4,8)



Answers

Find an equation of the tangent line to the parabola at the given point. $$x^{2}=2 y,(4,8)$$

Suppose you want to find the equation of tangent lines of the curve, X squared plus X, Y plus y squared equals four at the 40.20 So the first thing we have to do is to find the slope of the tangent line which is just the derivative of the curve evaluated at The given .20. Now by implicit differentiation we have derivative of X squared plus X, Y plus y squared. This is equal to the derivative of four. And so we have For the first term the derivative is two x plus product rule for the second term. Can we get um X plus or X times Dy over dx plus Why Times The Derivative of X? Which is one This plus the derivative of Y squared which is to Y times dy dx. And then this is equal to the derivative of four which is zero. Simplifying this, we get two X plus x dy dx plus Y plus two White Times. Dy DX. This equal to zero. Now combining all those with dy dx we have X plus two Y times do I. D X. And those without the I. D. S. On the other side we have minus two X minus y. Now dividing this by express to I've yet dy over dx which is equal to the negative of two, X plus Y. All over X plus two Y. And so at the point 20 we have the slope of the tangent line which is equal to dy over dx evaluated at 20 This gives us negative of two times two plus Y. Which is zero. That's all over. Exorcist two plus two times 0. Now we have Negative for over two or negative too. And so by point slope formula we have the equation of the tangent line which is equal to why minus ways of one equals m times x minus X of one. In which case we have X one, Y 1 equal to 20 and M equals negative two. So we have Why 0? That's equal to the slope, which is negative two Times X -2. And simplifying this, we get Y equals negative two X plus four. And so this is the equation of the tangent line.

Okay so you want to find the tangent of this car at this point? So first we're going to find the driven. So here we have two X plus two times Y plus X Y. Prime minus two. White by prime. Last one is equal to zero. So simplify this. Mhm. People. So we want to get My crime by itself. So we have our Prime -2. Why? Why I You see 4 to Uh huh. two X -2 Y -1. And I don't want we can't throughout my point. So you get two x minus two White Superintendent two x minus two Y minus one. That means that while prime sequences -2 X -2. Right. And that's well over two. X. managed to one. Mhm. So this is just the primitive but is now what we're looking for. So I'm not going to stop here. So we evaluate this at the right fancy you want to find this rope at this point? So that means Who have negative two times 1 -2 times two. And it's one divided by two times 1 -2 times two. So you have maybe 2: -4 -1. Two minus -7 over negative two Which you support with 7/2. Okay. Okay. You know. Okay so that's the slope. So and the sequel to this and now we just need the points for formula he says why minus do I value manage this? Because the slope times X minus this body. So now I used to The slope this event over two and its values for. So you sacrificed this CY -2 is equal to seven my less, she said, called Sir. Well I said 1-7 -7. 2 Plus two. What's this mean? This means that don't I was made. Perhaps so? This is the attentions. Yeah.

Hi, this is Claire. So was section 3.3, number 34 of Stuart Spot Calculus book. So we're gonna find the equation of the 10 gel on using dysfunction. That's to go forth minus two X squared, minus acts with the corn. It's 12 So this is the, um when your function the form for, um, the linear function where a represents slope so we're gonna do is we're gonna find the derivative, um, of the original equation. So that gives us or that's cute, Highness bar X, earnest one. And then we're gonna plug and X equals one, and then we get seven, which represents our slope A. We're gonna plug in seven to our, um, linear function. Why equals a explicit E? Why equals seven X must be plugged in corn. It's 12 for X and y you can't be be you called to negative five. And then our equation of the tension is like equal seven X minus 59 box or answer when we're done

We have the equation X squared plus two x y minus y squared plus X equals two. And we want to find the equation of the tangent line to the graph of this equation at the point one to using implicit differentiation. Now the first thing we should check is that the 0.12 is actually on the graph of dysfunction. So if we plug in X equals one and y equals two, we get one squared is one plus four is five minus four is one plus one is two. So it is actually on the graph of dysfunction. So we want to find the equation of a tangible it for the equation of a line. It's sufficient to know a point that that line goes through which we have. It's 112 and the slope of the line, and the slope of the line in this case is going to be Do you? Why over DX? And so to find this using implicit differentiation, we're going to take the derivative of both sides with respect to X, noting that why is a function of X so starting on the left, we get two x plus, and now we're going to use the product rule for derivatives that state that halftimes G prime is f prime G plus f g prime. So in this case, our first function is to X. The derivative is too most Bye bye. Why waas two x times theory votive of y with respect X, which is d wide the x minus. Now we want to take the derivative of y squared which by the chain rule we start by taking the derivative of the outer layer which isn't taking your inner layer. Why? And squaring it by the power rule. The derivative of that is to why then multiplied by the derivative of the inner layer. Why, which is do I d X finally plus the derivative of X is one equals the derivative of two is zero. So now we want to solved for d Y over DX. So I'm going to move all the d Y over. I guess I'll move. I want toe have all the terms that have a factor of d Y over DX on one side. So I'll move everything else to the right. So I have two X, do you? Why? Over the X minus two. Why d y over? DX equals minus two X minus two. Why? Minus one? Now? I'm going to factor out the D Y over DX from the left hand side to get to X minus two. Why? And right. So the final step to find D Y over DX is to divide by two X minus two. Why that gets us that d y over. DX equals minus two X minus two. Why? Minus one over to X minus two. Why? So what is the derivative D y over DX at the point X equals one, and why equals to? Well, we just plug in one and two. So d y over DX at 12 is going to be. Let's see. Minus two minus four is minus six minus one is minus seven. Divided by two times one is two minus four is minus two. So seven halves. Now we're going to We have the derivative, the slope of the tangelo in and a point that it passes through. We're going to use one of these standard formulas for a line. Why minus y one equals m X minus X one where m is the slope of the line and X one y one is a point on the line. So by plugging in the relevant information we get why minus two equal seven have X minus one and we're done.


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