Use implicit differentiation to find dy/dx of a curve with equation x^3 + yx^2 = y^2 + 1.

Begin by differentiating each term w.r.t x: d/dx(x^3) + d/dx(yx^2) = d/dx(y^2) + d/dx(1). the terms x^3 and 1 are simple enough to start of with: d/dx(x^3) = 3x^2 and d/dx(1) = 0. Next use the chain rule for the term y^2: d/dx(y^2) = d/dy(y^2) * dy/dx = (2y)dy/dx For the last term, yx^2, we differentiate using the product rule: d/dx(yx^2) = x^2(d/dx)y + y(d/dx)x^2 = 2xy + x^2(dy/dx) (Note that for y(d/dx)x^2 we use the chain rule again). Therefore with all terms differentiated we have: 3x^2 + 2xy + x^2(dy/dx) = (2y)dy/dx. Now we have to rearrange to get dy/dx: (2y - x^2)(dy/dx) =  3x^2 + 2xy ===> dy/dx = (3x^2 + 2xy)/(2y - x^2)