Implicit differentiation homework

[Hypochondriac]

Question:

Find the co ords of the turning points of y^3 + 3xy^2 - x^3 = 3

Attempt:

(differentiate w.r.t. x)
d/dx(y^3 + 3xy^2 - x^3) = d3/dx

3y^2(dy/dx) + 3(2xy(dy/dx) + y^2) - 3x^2 = 0

(divide through by 3)

y^2(dy/dx) + 2xy(dy/dx) = x^2 - y^2

(take dy/dx as a comon factor)

dy/dx(y^2 + 2xy) = x^2 - y^2

dy/dx = (x^2 - y^2)/(y^2 + 2xy)

now to find turning points, you set dy/dx = 0
but with explicit differentiation, there's only one variable, but there's 2 here, so I am stuck :/

 

[HallsofIvy]

"Turning points" must occur where dy/dx= 0 (though not all such points are turning points.

If dy/dx = (x^2 - y^2)/(y^2 + 2xy) and y^3 + 3xy^2 - x^3 = 3, can you solve those two equations for x and y?