A curve C has equation: x^3+2xy-x-y^3-20=0. Find dy/dx in terms of x and y.

First we need to make sure we understand implicit differentiation. As we are differentiating with respect to x, y has to be treated differently, this is because it could be anything from a constant to a function of x say f(x). Thus we don't know what its derivative with respect to x is but we do know how to represent it; as dy/dx. So to answer this question we will use the product rule along with what I have just described. For instance take the 2xy term, this will give an implicit differentiation of 2y +2x(dy/dx). Using this idea we can differentiate the original equation term by term to get 3x^2+2y+2x(dy/dx)-1-3y^2(dy/dx)=0. Isolate the (dy/dx) terms to get (2x-3y^2)(dy/dx)=1-2y-3x^2. Divide through (2x-3y^2) to get (dy/dx)=(1-2y-3x^2)/(2x-3y^2) which is the final answer.