Note first that by the chain rule d/dx(y^4)=d/dy(y^4) times dy/dx=4y^3dy/dx
Also note a term like x^2y is a product and will require the product rule to differentiate it
In your first question differentiate each term (BOTH sides) with respect to x
Get 4x^3+(8xy+4x^2dy/dx)+4y^3dy/dx=0 The part in brackets came from using the product rule)
So (4x^2+4y^3)dy/dx=-4x^3-8xy dy/dx=-4(x^3+2xy)/4(x^2+y^3)=-(x^3+2xy)/(x^2+y^3)
The process wew have used here is called implicit differentiation.