# Implicit Differentiation

1. Nov 1, 2009

### temaire

1. The problem statement, all variables and given/known data
Find all the points on the curve $$x^{2}y^{2}+xy=2$$ where the slope of the tangent line is -1.

3. 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.

Last edited: Nov 1, 2009
2. Nov 1, 2009

### Staff: Mentor

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

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.

3. Nov 1, 2009

### temaire

Thanks, I got it.