Finding Points on a Curve with Tangent Line Slope -1

[temaire]

Homework Statement

Find all the points on the curve $$x^{2}y^{2}+xy=2$$ where the slope of the tangent line is -1.

The Attempt at a Solution

I differentiated both sides of the equation and got:
$$\frac{dy}{dx}=\frac{-2xy^{2}-y}{x^{2}2y+x}$$

I know that $$\frac{dy}{dx}=-1$$, but if I substitute -1 in, I won't be able to go any further since I have two unknown variables. I would appreciate any help.

 

[Mark44]

$$\frac{dy}{dx}=\frac{-2xy^{2}-y}{2x^{2}y+x}~=~\frac{-y(2xy + 1)}{x(2xy + 1)}$$

As long as 2xy + 1 $\neq$ 0, you can cancel the factors of 2xy + 1, leaving a much simpler derivative.

Also, you want to solve the equation dy/dx = -1, not dy/dx = 1, as you had. Notice that you still have two variables, but all that means is that there are lots of solutions.

 

[temaire]

Thanks, I got it.