The “egg carton” function is a good example for connecting partial derivatives to contour plots, since it has a richly varying surface.

Plot3D[ Sin[x]*Cos[y], {x,-2*Pi, 2*Pi}, {y,-2*Pi, 2*Pi} ]

ContourPlot[ Sin[x]*Cos[y], {x,-2*Pi, 2*Pi}, {y,-2*Pi, 2*Pi}, Contours->20 ]

The familiar “D” function actually takes partial derivatives.

t = Sqrt[ u^2 + v^2 + w^2 ]

D[ t, u ]

D[ t, v ]

D[ t, w ]