In Exercises 29 and 30, find the slope of the curve at the giv...

Question

In Exercises 29 and 30, find the slope of the curve at the given points.

(x² + y²)² = (x – y)² at (1,0) and (1,–1)

Accepted Answer

Hello, in this video, we are going to be finding the slope to the tangent line of the given curve. The curve given to us is 3 X2 + 2 Y2. that quantity raised to the power of 2, equal to 9, multiplied by X + Y raised the power of 2, and we want to find the slope of the tangent line at the point -1.0. So, in order to find the tangent line, the first thing we, or at least the slope of the tangent line, the first thing we need to do is we need to take the derivative of this function. Notice that this is an entire function in terms of both X and Y. So, in order to approach this problem, we are going to take the derivative with respect to X by using implicit differentiation. On the left hand side, we will go ahead and use the chain rule. The derivative of the left hand side will be 2 multiplied by 3 X2 + 2 Y 2, then multiplied by the by the derivative of the innermost functions, which will be 6 X plus 4 Y. Now, since we are taking a derivative of Y with respect to X, we are going to end up generating a DYDX term, and this is going to be the derivative of the left hand side. On the right hand side, we will also use chain rule. We will end up with 18, multiplied by the quantity X plus Y multiplied by the derivative of the innermost function, which is 1 + 1, but again, derivative of Y is going to be DYDX, so we have 1 plus DYDX. Now we need to simplify this function. We are going to foil the left hand quantity and the right hand quantity together first. That is going to leave us with 2 multiplied by 18 X cubed plus 12 X 2 Y DYDX. Plus 12 x y squared. Plus 8 Y cubed DYDX, and this will equal to 18 multiplied by X plus XDYDX plus Y plus YDYD X. Now what we are going to do is we're going to distribute 2 on the left hand side and 18 on the right hand side. That is going to leave us with 36 X cubed plus 24. X squared Y D Y D X Plus 24 XY squared, plus 16 Y cubed DYDX. And that is equal to 18 X plus 18 X DYDX plus 18 Y plus 18YDYDX. Now, since our goal is to solve for the derivative DYDX, what we want to do is we want to get all the terms with DYDX to the left hand side of the equation, and all the constant terms to the right-hand side. We have 4 terms in total that have a DYDX differential. So we're going to move all of these to the left hand side and the rest to the right hand side. That is going to leave us with 24 X squared Y DYDX plus 16 Y cubed DYDX. Minus 18 XDYDX. Minus 18 Y DYDX, and this is going to equal to 18X plus 18 Y. Minus 36 x cubed minus 24 X Y squared. That was a lot of terms to keep up with, but notice that every term on the left-hand side of this equation now has a DYDX multiple. So what we can go ahead and do is we can factor out a DYDX from the left-hand side of this equation. That will leave us with DYDX multiplied by 24 X 2 Y plus 16 Y cubed minus 18 X. Minus 18 Y, and this is equal to 18 X plus 18 Y minus 36 X cubed minus 24 XY squared. And finally, to isolate DYDX, we will just go ahead and divide by that quantity on the left hand side, and this will leave us with DYDX is equal to 18X plus 18 Y. Minus 36 X cubed, minus 24 XY squared. And this will all be divided. By 24 X 2 Y plus 16 Y cubed minus 18 X minus 18 Y. So this is going to be the derivative that gives us the slope to the tangent line. Now, again, earlier from the problem, we are trying to find the slope of the tangent line at the point -1.0. So what we are going to do is we are going to plug in the value -1.0 into this derivative. This means that for every X, we are going to plug in -1, and for every Y, we are going to plug in 0. Now, if we plug in 0 for Y, any term that has a Y in our derivative is just going to become zero. So these are going to be the following terms that cancel out because they contain a Y variable. Now what we are left with is X being -1. If we plug in -1, 18 times -1 will be -18. -1 to be -1 and -36 multiplied by -1 is 36. This will then be divided by -18, multiplied by -1, which is 18. In the numerator, 18 + 18 + 36 gives us 18 and 18 divided by 18 is going to be 1. So the slope to the tangent line at -1.0 is going to be 1, and this is the solution to the problem. So I hope this video helps you with the implicit differentiation, and I will go ahead and see you all in the next video.

In Exercises 29 and 30, find the slope of the curve at the given points.

(x² + y²)² = (x – y)² at (1,0) and (1,–1)

Key Concepts

Implicit Differentiation

Chain Rule

Evaluating Derivatives at Specific Points

Watch next