If you partially differentiate the equation of a hyperbola w.r.t. x or y do you get the equation of its asymptotes? I know that if you do partially differentiate it, the two lines that you get, intersect at its center. This is true for any conic and pair of straight lines. What about other conics? I.e, if you partially differentiate the equations for a parabola, ellipse, circle, what exactly do you get?