Plots of relations and implicit differentiation

Mathematica can plot relations as well as graphs of functions

ContourPlot[((((x^2)+(y^2)))^2)==2 x y,{x,-1,1},{y,-1,1},
Axes-> True]

A relation discussed in the text under the topic of implicit differentiation.

Here is the relation:

eqn = (x^3 + y^3 == 3 x y)

Here is the plot of the relation

ContourPlot[x^3 + y^3 == 3 x y,{x,-2,2},{y,-2,2},
Axes-> True]

Find the points on the relation, for a given value of x

Solve[{x^3 + y^3 == 3 x y, x == .5}, y]

Implicit differentiation of the relation

eqn1 = eqn/. y -> y[x]
D[eqn1,x]
Solve[%, y'[x]]

Find the derivative of the implicit function when x = .5, y = -1.264458978

deriv =  (((x^2)-y[x])/(x-((y[x])^2)))
deriv/. {x -> .5, y[x]->-1.264458978647181`}