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Differentiate implicitly and then determine the equation of the tangent line to the curve defined by the given equation at the indicated point.$$y^{4}-2 x^{2} \ln y=-8 x+4 x^{2} y^{3}+y(2,1)$$

$$y=-\frac{8}{53} x+\frac{69}{53}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

Campbell University

McMaster University

Idaho State University

Lectures

03:43

Differentiate implicitly a…

03:10

Compute the slope of the t…

01:35

Find an equation of the ta…

02:08

Find the equation of the t…

02:14

Find an equation of the st…

02:45

Find $y^{\prime}$ and the …

01:27

02:24

Find the tangent to the cu…

The underlying theme in this problem is the tangent line. And to find the equation a tangent line, you need the alright down point first because they give you the ordered pair and the problem eyes to one. So we already have half the information that we need. What we have to find is a slope and we find the slope by doing the derivative D u i d x at the order pair to one eso Now we can look at the problem. Why do the fourth minus two X, uh, squared natural log of y minus eight x What's four x squared y cube is gonna get ugly. Plus why so starting with the derivative You know, four y cube d y d x But if you notice the next term has a product in here. So it's the product rule where the derivative of X squared is to exile. Make that four x leave natural log of y alone. Now you leave the two x squared alone in the derivative of natural log of y is one of the y t y DX equals on the right side, the derivative of negative A tips negative. We have another product rule, so that would be eight x y cute plus three times forward 12 X squared y squared D Y DX on the derivative of Why is D Y d exits one d y DX, but I've run out of room. So now, because this is already ugly enough. I didn't do a second line is I'm unplugging one. And for all my wise to and for all of my exes and well, you'll notice is one cubed is still one times for us. Four b. Y the x Now I'm not gonna worry about this X because I know natural log of one is zero. So that's that's going away. Aan den two squared us four times that to be negative. Eight one divided by one is one. So we have eight d Y d. X there, and on the right side we have negative eight. Let's see eight times. Two times one will be 16 C two squared is four times 1 12 times four is 48 de y dx les de y dx. So from here, what I can do is I can solve by getting my like terms on the same side. I'd be subtracting negative. 48 d Y d x subtracting one. Do I. D. X onto the left side and on the right side. I could just combine these together. It'll be eight. So four negative four. Native 52 53 and it's kind of a nobly answer, but it does match the answer key. So I know I'm doing this right. That the slope at that point. Now I know I'm writing on Lee D Y D X, but it's on. Lee happens at the ordered pair to one eyes eight over negative 53. So now we have enough to answer a question because our equation of the Tanja lines, why it was a slope and then x minus R X coordinate plus the y coordinate would go back to the point. The X coordinate was, too. The wide coordinate was one from at the point that was given. And then we found the slope is negative. 8/53 and this is our final answer. You can simplify if you want, but that's gonna be busy work

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