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

S=Plot3D[f[x,y],{x,-1,1},{y,-1,1},BoxRatios->{2,2,1},Mesh->3,Axes->False,Boxed->False(*,PlotStyle->Opacity[0.4]*)]

fx[x_,y_]=D[f[x,y],x]

fy[x_,y_]=D[f[x,y],y]

P=Plot3D[fx[1/2,1/2](x-1/2)+fy[1/2,1/2](y-1/2)+f[1/2,1/2],{x,-1,1},{y,-1,1},PlotRange->{0,1},ClippingStyle->None,Mesh->None,PlotStyle->Opacity[0.4]]

A=Graphics3D[Style[{{Red,Ball[{1/2,1/2,Sqrt[2]/2},0.03]},Text["A",{0.6,0.5,Sqrt[2]/2}+0.1]},Large]]

Show[S,P,A]

fx[1/2,1/2]

fx[1/2,1/2](x-1/2)+fy[1/2,1/2](y-1/2)+f[1/2,1/2]

Simplify[%]