45–50. Tangent lines Carry out the following steps.

Question

b. Determine an equation of the line tangent to the curve at the given point.

(x²+y²)²=25/4 xy²; (1, 2)

Accepted Answer

Hello. In this video, we are going to be calculating a tangent line. So the problem tells us to find the equation of the tangent line to the curve defined by the following equation, at the point 2.2. Now, the equation given to us is the quantity 3X2 + 4 Y 2, that entire quantity squared equal to 98 XY 2. So, since we are asked to find the tangent line to this equation, what we need are two pieces of information. First, we need a given point, which is going to be 2.2, but then we are going to need the slope at the point 2.2. In order to find the slope at 22, what we need to do is we need to take the derivative of our given equation. And we will be taking the derivative with respect to X. Now notice that our equation has variables of both X and Y. What this means is that we are going to have to take the derivative using implicit differentiation. Now, this is a slightly long problem, so let's go ahead and break this down piece by piece. Let's start by first taking the derivative of the left hand side of the equation. Now again, the left hand side is given to us as a quantity raced to the 2nd power. Since we have a function within another function, what this means is that we are going to have to take the derivative using the chain rule. So we will take the derivative of the outermost function, which is going to be the exponent of 2. By the power rule, that is going to give us 2 multiplied by the original inside function, which is 3 X2 + 4 Y squad. Then we will need to multiply this quantity by the derivative of the innermost function, 3 X2 + 4 Y2. Now, the derivative of 3 X2 is going to be 3 multiplied by 2 X. That will be the derivative of X squared by the power rule. Now, we will do the same thing with 4 Y squared. The derivative of that term will be 4 multiplied by the derivative of Y2d, that will be 2 Y. But realize that the derivative of Y2d with respect to X is going to generate DYDX through implicit differentiation. This is going to be the derivative of the innermost function. Now what we need to do is we need to simplify. So, we have 2 multiplied by 3 X2 + 4 Y squared. Now, 3 multiplied by 2X is going to give us 6 X, and 4 multiplied by 2 YDYDX is going to give us 8 YDYDX. Now what we need to do is we need to foil these two quantities together. So, by foiling the two quantities together, we will get 2 multiplied by 18 X cubed, plus 24 X 2 Y DYDX. Plus 24 XY squared. Plus 32 Y cubed. D Y D X. And finally, what we need to do is we need to distribute 2 across this entire quantity. Distributing 2 will give us a final simplified form of the left-hand side, as 36 X cubed plus 48 X squared YDYDX. Plus 48 XY squared. Plus 32, I'm sorry, not 32, 64. Y cubed D Y D X. This is going to be the derivative of the left hand side of the equation. Now what we want to do is we want to take the derivative of the right-hand side. Now, again, the right-hand side of the equation is given to us as 98 XY squared. In order to take the derivative, we are going to have to use the product rule. So, what we will do is we will take the coefficient 98 and multiply this by the product rule of XY2. This will give us the following. It'll be X multiplied by the derivative of Y2d. Now, the derivative of Y2d will be 2 Y by the power rule, but again, since we are taking the derivative of Y with respect to X, we will generate a DYDX term through implicit differentiation. Then we will add Y2d and multiply this by the derivative of X, which will just be one through the power rule. Now we need to simplify. X multiplied by 2 YDYDX is going to give us 2 XYDYDX. So we have 98 multiplied by 2 XY, DYDX plus Y squared. Then we will distribute 98 across this quantity, and the simplified derivative of the right-hand side of the equation will be 98s multiplied by 2 XY, which is 196 XYD YDX plus 98 Y squared. Now that we have the derivatives to both sides of the equation, we are going to set both sides equal to each other. Now, that is going to be 36 X cubed plus 48 X 2 YDYDX. Plus 48 XY squared, plus 64 Y cubed DYDX, and that will equal to 196 X Y D YDX plus 98 Y squared. So this is quite a lengthy derivative, but what we want to do now is we want to solve for the derivative DYDX. In order to do this, we will take every term that involves a DYDX term and move it to the left-hand side of the equation. And any term that does not have a factor of DYDX will be moved to the right hand side of the equation. That will allow us to rewrite the equation as 48 X 2 Y. DYDX plus 64 Y cubed DYDX minus 196 XYDYDX and that will equal to 98 Y2 minus 36 X cubed minus 48 XY squared. Now what we want to do is you want to factor out a DYDX from the left hand side of the equation. That will allow us to rewrite the left hand side. As DYDX multiplied by 48 X 2 Y plus 64 Y cubed minus 196 XY. Now, what we want to do in order to solve for DYDX is divide everything by the quantity 48 X X 2 Y plus 64 Y cubed minus 196 XY. That is going to give us DYDX equal to 98 Y squared. Minus 36 x cubed. -48. X Y squad. Divided By 48 X 2 Y plus 64 Y cubed minus 196 XY. So what we have now is the derivative to the given equation through implicit differentiation. Now what we want to do is we want to solve for the slope at the point. At the given point, which is 2.2. If we plug in 2.2 into this derivative, that is going to give us a slope equal to -5 divided by 2. So now what we have is the slope located at 2.2. What we can do now is we can solve for the tangent line. That is going to give us Y minus 2 equal to the slope, -5 divided by 2 multiplied by the quantity X minus 2. We will distribute -5 divided by 2 across X minus 2, leaving us with Y minus 2 equal to -5 divided by 2 X plus 5. And if we add 2 to both sides of the equation, the final form of the tangent line is going to be Y is equal to -5 divided by 2 X plus 7. This is going to be the equation to the tangent line at the point 2.2 for the given equation. And with that being said, the overall solution to this problem is going to be B. So, I hope this video helps you in understanding on how to find the tangent line, and I will go ahead and see you all in the next video.

45–50. Tangent lines Carry out the following steps. <IMAGE>
b. Determine an equation of the line tangent to the curve at the given point.
(x²+y²)²=25/4 xy²; (1, 2)

Key Concepts

Implicit Differentiation

Tangent Line Equation

Chain Rule

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