Question

Find dy/dx by implicit differentiation x^3+3x^2y+y^3=8

Answer 1

Just remember, for y, just leave the dy/dx there

3x²+3(2xy+x²dy/dx)+3y²dy/dx=0
3x²+6xy+3x²dy/dx+3y²dy/dx=0
minus (3x²+6xy) from both sides
3x²dy/dx+3y²dy/dx=-3x²-6xy
undistribute dy/dx
dy/dx(3x²+3y²)=-3x-3xy
divide both sides by (3x²+3y²)
$\frac{dy}{dx} = \frac{-3x^2-6xy}{3x^2+3y^2}$
$\frac{dy}{dx} = \frac{-x^2-2xy}{x^2+y^2}$