Have you studied "conic sections" before?
http://math2.org/math/algebra/conics.htm
The glass you show is at least a little bit like part of a cone (on the sidewalls at least). So, taking your partial derivatives with respect to y is like making a conic section by dropping a vertical plane thru the cone. This gives a hyperbolic-like shape (perhaps not a perfect hyperbola, since the glass is not a perfect cone). There is no problem finding slopes/tangents on these types of curves.
Hopefully, this gives you the tool to visualize it now.