Each y is some (unknown) function of x, so
to differentiate y^3 and -3y^2 we will use the power rule and the chain rule (a combination sometimes called 'the generalized power rule')
To differentiate x^2y we'll need the product rule.
(I use the order: (fg)' = f'g+fg' and I'll put the product in brackets [ . . ])
d/dx(y^3+x^2y+x^2-3y^2) = d/dx(y^3)+[d/dx(x^2y)]+d/dx(x^2)-d/dx(3y^2)
color(white)"sssssss" = 3y^2 dy/dx +[ 2xy+x^2 dy/dx] +2x -6y dy/dx
color(white)"sssssss" = (2xy +2x) + (3y^2 +x^2 -6y )dy/dx
That's all we can do. If we had an equation, we could solve for dy/dx.