x^2 + y^2 = (xy - 2)(x + y)

x^2 + y^2 = x^2 y + xy^2 - 2x - 2y

Now take the derivative:

2x + 2yy' = 2xy + x^2 y' + y^2 + 2xyy' - 2 - 2y'

2yy' - x^2 y' - 2xyy' + 2y' = -2x + 2xy + y^2 - 2

y'(2y - x^2 - 2xy + 2) = y^2 + 2xy - 2x - 2

y' = (y^2 + 2xy - 2x - 2)/(2y - x^2 - 2xy + 2)

Now, we are at the point (-1, -1):

y' = ((-1)^2 + 2(-1)(-1) - 2(-1) - 2)/(2(-1) - (-1)^2 - 2(-1)(-1) + 2)

y' = (1 + 2 + 2 - 2)/(-2 - 1 - 2 + 2)

y' = 3/-3 = -1

-Dan