Compute the Gauss map and its derivative for the cone parameterized as follows
X(u,v) =(vcosu, vsinu, v)
What does the image of the Gauss map on the sphere look like?
Xu = ( -vsinu, vcosu, 0 )
Xv = (cosu, sinu, 1 )
So the normal vector N = Xu x Xv = v(cosu, sinu, -1), the unit normal is n=(cosu, sinu, -1)/\sqrt{2}, and it is the Gauss map G. Its image is a circle on the unit sphere.
The derivative at p=X(u,v) is a linear map dG from tangent space Xp to the tangent space of the unit sphere S_G(p) such that:
dG(Xu) = (-sinu, cosu, 0)/\sqrt{2}
dG(Xv) = 0