Partial Derivatives

We define F to be a function of two variables whose graph lies on the unit sphere:

F[x_, y_] := Sqrt[1 - x^2 - y^2]

F[x, y]

Here are its partial derivatives

D[F[x, y], x]
D[F[x, y], y]

We define functions representing its partial derivatives:

Fx[x_, 
        y_] := -(x/(Sqrt[1 - (x^2) - (y^2)]))
Fy[x_, y_] := -(y/(Sqrt[1 - (x^2) - (y^2)]))

Here is the graph:

p1 = 
      Plot3D[F[x, y], {x, -Sqrt[2]/2, Sqrt[2]/2}, {y, -Sqrt[2]/2, 
          Sqrt[2]/2}, AspectRatio -> Automatic]

(This is code to draw the tangent vectors obtained from the partial derivatives)

lx[x_, y_] := 
      Graphics3D[{RGBColor[1, 0, 0], Thickness[ .01], 
          Line[{{x, y, 
                F[x, y]}, {x, y, F[x, y]} + {1, 0, Fx[x, y]}/
                  2}]}]
ly[x_, y_] := 
      Graphics3D[{RGBColor[0, 1, 0], Thickness[ .01], 
          Line[{{x, y, 
                F[x, y]}, {x, y, F[x, y]} + {0, 1, Fy[x, y]}/
                  2}]}]

TangentVector[x_, y_] := {p1, lx[x, y], ly[x, y]}

This shows the tangent vectors <1,0,Fx[x,y]> in red and <0,1,Fy[x,y]> based at the point (x,y)=(0,0)

Show[TangentVector[0, 0]]

at (x, y)=(0.4, 0)

Show[TangentVector[ .4, 0]]

At (x, y)=(0.5, 0.5)

Show[TangentVector[ .5,  .5], 
      ViewPoint -> {2, - .5,  .5}]