Partial Derivatives after a change in coordinates

[Endo]

I have a surface in R3 defined as a differentialable height map f(x,y) over a plane of two orthogonal basis vectors u = (x1, y1, z1) and v = (x2, y2, z2). I then find a point P(x,y,z = f(x,y)) on the surface, and want to set up my coordinate system such that the normal at P is now the Z-axis, and the first basis vector is determined somehow (and so the second basis vector can be determined by crossing the normal with the new x basis vector).

I want to determine the partial derivatives of the surface at that point in the new coordinate system (i.e. Hxx, Hxy, Hyx, Hyy), but I'm not entirely sure what transformation I need to do to get the surface defined in the new coordinate system.

Any clues?