HW4: Problem 12Previous ProblemProblem ListNext Problempoint) Use implicit differentiation to find the slope of the tangent line to the curve defined by zy8 + 2xy 3...

In Exercises $29-44,$ solve the given problems by using implicit differentiation. Find the slope of a line tangent to the curve of the implicit function $x y+y^{2}+2=0$ at the point $(-3,1) .$ Use the derivative evaluation feature of a calculator to check your result.

We want to find the slope of the tangent line of the curve, Y equals to co tangent. Yeah. A three X. Where x equals pi over 12. So we see that when we take the derivative of this, end up getting this graph right here. Um And we want to remember that when we take the derivative of the code tangent of X, it's going to be negative Costa Rican X. A negative coast again squared X. So because we get negative Costa Rican spread act, it makes sense that this is the graph that we end up with. So we're going to evaluate this With x equals high over 12. So we see that we end up getting a negative 12, which means that is the slope of our graph. Um at Pi over 12 where F a bex two eagles to co change in three exes are function.

Oh we want to find the slope of the tangent line to the 0.11 on the equation. X cubed plus two X Y plus y squared equals support to solve this problem, we're going to use the process of implicit differentiation which from which we differentiate both sides with respect to X. And solve for dy dx the rate of change of the equation as we listen to steps here. Then we'll find dy dx at that point to find the slope. So first differentiate every term with respect to X in this equation. This gives us D D X X plus two xy xy squared. Because deedee export on the left hand side we take a simple derivative followed by the product rule followed by another simple derivative, unpleasant generalization to obtain three X squared plus two Y plus two X. Dy dx plus two Y. Dy dx on the right hand side seats for the concert and become zero. Next we saw for dy dx. Let's isolate dy dx on one side of the equation. Notice how I made this a little bit simpler here. So do I D X two X plus two Y equals negative three X plus two Y. Or do I d x equals negative three X plus two Y over two X plus two Y. Next to find the slope. We plug in the 20.2. Dy dx +11 equals negative three plus 2/2 plus two. Or negative 5/4.

Okay. This question is asked us to find the slope of the tangent line with this equation at X equals 0.15 And then verify it using the derivative finding function on the calculated. Okay, so we need to take the derivative clearly, we're going to use quotation role. So why prime equals X. The denominator times the derivative of the top. The derivative of the top will be six. Co sign three X. That three popping out in front minus the derivative. The bottom which is one times the original top which is to sign three X. All over the denominator squared. Ok. We don't have to worry about simplifying this down too much because we're substituting a number in here. We're substituting in that 0.15 15 times three will be a 30.45 Make a little bit easier calculations. Okay, substituting in 0.15 Sorry about that everywhere we see an X. Yeah, we're dividing by 0.5 square. Okay, we use that calculator for this number crunching So set up a fraction 0.15 six co sign 0.45 Okay, my ass to sign 0.45 divided by 0.15 squared gives us negative 2646 Oh. This is the slope of the tangent line. We can just check that. Using the derivative. Finding function on a calculator. Okay, on the T I 84 I'll show you where that is. Second. Okay, we're gonna go into math, we're gonna go down here to end the riff. We'll find us a numerical derivative. We're looking for DDX we're gonna put our function in here. We'll set it up as a fraction is to sign three X over X. And we wanted the numerical derivative at 0.15 So we put that in here and there, it confirms it for us. So, slope of the tangent line at that point negative 2.646