Use implicit differentiation to find dv for the equation v =2-"+4Find the equation to the tangent line to this curve at (-1,2)....

Find an equation for the tangent line to the curve at the given point. Then sketch the curve and tangent line together.
$y=2 \sqrt{x}, \quad(1,2)$

In this problem rescues and present the transition to find the question of defensive line to the given tour at a given 0.1 Health. We know the question of one will be up for y. Minus or not is director of dysfunction of a little argument point? Excellent times explains. It's not so in order to find the question the personally to pack light or find derivative related given 40 So to do that last time David off all the perfect suspect X we have to expose to wine times why prime that is equal to two times two x ray plus, um, to Weisberg when it's X times. Now there was a look in your function, which is four X plus four. Why times y prime minus one. All right. In order to find directive at people that that given point Les Plus X not on a warning in. So we have zero plus one crime that is equal to two times zero plus one. Hell, uh, my zero. That's like by zero plus two. Y pine likes one from this. We see that Why prime is equal. One off course. This is they were to ever let that given 10.0. Now, since we know this less like what your envy have Wyman's health is a photo one Times X minus zero Promise me that when the question of friends let s x plus one.

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.

This question asks us to use impressive differentiation to figure out what the drift of this and on the tension line. Okay, we're gonna be using the Jame rule on the term Why squared and then what's in parentheses to X squared Plus two. I scored minus X Prince C squared two acts is the derivative of X squared plus two wise, the derivative a y squared times. Why prime make the expert at the new proficient What's in parentheses stays the same and take the derivative to X squared. The derivative is for acts derivative of two. I squared is for why? Why prime minus one? Because the derivative of negative X is negative one not when X's year on Why is 1/2 this means we can simply pull again plug in the values x and y into what we just figured out. This gives us why prime is two times 1/2 times to y prime minus one which is equivalent to simply one is why prime. Okay, now use point slope form. We're looking at the 0.0.0 1/2. Therefore, why minus 1/2 is one times X minus zero, which means why is simply X plus 1/2

Ms Problem. Where's to use instant differentiation? Finding question, going to line up a teacher and given point to one. All right, you know that the question of tension lines will be your wine one. So I'm not busy with territorial dysfunctional bullet at a given point. What Spike by X minus X Not where? Why not? Is one the next, not is two. So we need to point their bitter in order to do that. Less activity or both sides With respect, X, we have two X plus or diss term will be using protocol to one plus two extends for crime plus age White on white crime Z zero. Now, let's pluck given, uh, eggs and one in order to fund derivative at this given points in order to find this term, um, we have two times two plus two times one plus two times two. Why prime plus a 10 4 times. Why? Prime is zero firmness. We see that that white prom should be negative. Six over 12 on. That is a negative one. Health. Now we have everything that you need. Then we can right? You crazy a pendant, Linus, Why? By this point which is. Why not? Times there were terminated one hell, Well, why buy X ones too homeless? We see that the question of standard line is that two minus X or two.