directional derivative

Manipulate[Show[Plot3D[x^2-y^2,{x,-4,4},{y,-4,4},AxesLabel->{x,y,z},ViewPoint->{1,1,1}],ContourPlot3D[y==Tan[T]*x,{x,-4,4},{y,-4,4},{z,-16,16}]],{T,0,2Pi}]

Example 1

Show[Plot3D[Cos[x]*Cos[y],{x,0,2Pi},{y,0,2Pi},AxesLabel->{x,y,z},ViewPoint->{1,1,1}],ContourPlot3D[y-Pi==3*(x-Pi/2),{x,0,2Pi},{y,0,2Pi},{z,-1,1}]]