Find the slope of the curve x³y³ + y² = x + y at the points (1...

Question

Find the slope of the curve x³y³ + y² = x + y at the points (1, 1) and (1, -1).

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the slope of the tangent to the curve. X2 Y2 + 2 XY e equals 5 X + 4 at the point for 1. A says it's -1/8, B and 8, C negative 8, and the D negative quarter. Now how can we find the slope of the tangent to the curve? Well, recall that the slope of the tangent to a curve is equal to the equal to the derivative of the curve at that point. So if we can differentiate the equation for our curve and then solve it for the value X to be 4 and the value y to be 1, the coordinates of our point, then we'll get the slope of the tangent to the curve. So first, let's start by differentiating the equation of the curve. So differentiating the equation. Of the curve, OK. Notice here that we'll have to use implicit differentiation, so we're differentiating it. Or rather, if we're differentiating it with respect to X, then the derivative of X 2 Y squared. Is going to be 2 XY 2 plus 2 X 2 Y D Y D X based on the the. Uh, product rule of differentiation. Next, when we differentiate 2 XY, it's going to be 2 Y plus 2 XDYDX, OK. The derivative of 5 X would be 5 and the derivative of 4 would be 0. Now we need to solve for DYDX. Now, in that case, if we can group our like terms together, OK. Then we're gonna have DUIDX. Multiplied by 2 X 2 Y plus 2 X. OK. And this is going to be equal to 5 minus 2 XY squared, OK, minus 2 Y. Thus, this means that the derivative. Is going to be equal to 5 minus 2 x y squared minus 2y. All divided by 2 x square y plus 2 X. Now that we have the derivative, then we can find a slope. Because now the slope. OK. At the point for 1, OK. Let's say that that slope is M. To find it, we're going to substitute X for 4 and Y41 into our derivative. So that's 5 minus 2 multiplied by 4 multiplied by 1 squared. Divided by 2 multiplied by 4 squared multiplied by 1 plus 2 multiplied by 4. I know when we evaluate here our quotient, then we should get it to be equal to negative 1/8. Therefore, the slope at the 0.41 of the curve is -18. If we look back on our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Find the slope of the curve x³y³ + y² = x + y at the points (1, 1) and (1, -1).

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