Applying the basic derivation rules such as the power rule, chain rule, and product rule,
Solving for dy/dx yields
The slope is vertical when the denominator of dy/dx is zero and the numerator is non-zero. It must be noted that all (x,y) that satisfy 3y^2 - 2x = 0 must also satisfy the original curve.
I did not want to attempt to solve the system of two equations (it would be quite difficult as well since it involves cubics), so I used Wolfram to find the points on the graph where dy/dx has denominator zero. WolframAlpha lists (0.653, -0.660) and (2.046, 1.168) as the only such points. The graph of the curve and the parabola 3y^2 - 2x = 0 can be found by going to the URL below (the intersection of the curves show the desired points):
FYI: Another alternative might be to use polar coordinates and then differentiate. However, this solution might be longer since you will have to solve for r first.
