This demo shows the slice curves at a point (P,f(P)), where P is a point in the domain.

There are three slice curves representing partial derivatives with respect to r, θ and z.
These slice curves are normal to a vertical cylinder of radius r at point P, tangent to a vertical cylinder of radius r with constant z at point P, and tanget to a vertical cylinder of radius r with constant θ respectively.

 The slopes of the slice curves at P are represented by the colorings of these slices, and give the partial derivatives with respect to r, θ and z at (P,f(P)). You can use the hotspot in the "Domain" window to get the partial derivatives of every point in the domain.