Pressure Height Dependence

Consider a horizontal infinitesimally thin disc with face area A and thickness dz submersed in a fluid.  The difference in the forces applied to the top and bottom faces of the thin disc is the weight of fluid with the same volume.  This weight is ρgAdz where ρ is the fluid density and g is the freefall acceleration.  Therefore, if dP is the infinitesimal change in pressure:

dP = -ρgdz