Implicit differentiation is basically done in cases where #y# cannot be explicitly written as a function of #x#.
In this case,
#-3=(x^2+y)^3-y^2x #
Differentiating both sides w.r.t. #x#
#=> -3(d(1))/dx = [d(x^2+y)^3]/dx - (d(y^2x))/dx#
Using chain rule to evaluate #[d(x^2+y)^3]/dx# & product rule to evaluate # (d(y^2x))/dx#
#=> 0 = [d(x^2+y)^3]/(d(x^2+y))*[d(x^2+y)]/dx - [y^2dx/dx + xdy^2/dx]#
#=> y^2dx/dx + xdy^2/dx = [d(x^2+y)^3]/(d(x^2+y))*[d(x^2+y)]/dx#
Using sum rule to evaluate #[d(x^2+y)]/dx# & chain rule to evaluate #dy^2/dx#
#=> y^2 + x*dy^2/dy*dy/dx = [3*(x^2+y)^2]*[dx^2/dx + dy/dx]#
#=> y^2 + x*2y*dy/dx = 3(x^2+y)^2*[2x+dy/dx]#
#=> y^2 + 2xy*dy/dx = 6x(x^2+y)^2 + 3(x^2+y)^2*dy/dx#
#=> 2xy*dy/dx - 3(x^2+y)^2*dy/dx = 6x(x^2+y)^2 - y^2 #
#=> [2xy - 3(x^2+y)^2]*dy/dx = 6x(x^2+y)^2 - y^2#
#=> color(red){dy/dx = [6x(x^2+y)^2 - y^2]/[2xy - 3(x^2+y)^2]}#
