Use implicit differentiation to find dy/dx.  We differentiate each x term as usual and when differentiating each y term, you add dy/dx since we are differentiating with respect to x.  Then isolate dy/dx to solve.

x^3 + y^3 - 3xy = 0

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

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

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

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

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