=================================================================

Manipulate[
 {{xmin, xmax}, {ymin, ymax}} = {{-2, 2}, {-2, 2}};
 isize = Medium;
 zmin = NMinimize[{f[x, y], {xmin <= x <= xmax, 
      ymin <= y <= ymax}}, {x, y}][[1]];
 zmax = NMaximize[{f[x, y], {xmin <= x <= xmax, 
      ymin <= y <= ymax}}, {x, y}][[1]];
 zlevel = -2(* zmin - 1.5 (zmax-zmin)*);
 Show[
  Plot3D[f[x, y], {x, xmin, xmax}, {y, ymin, ymax}, 
   ImageSize -> isize, PlotStyle -> Opacity[.5], 
   PlotRange -> {xmin, xmax}
   ],
  Graphics3D[{Red, Arrowheads[Large], PointSize[Large], 
    Thickness[Large]
    , Point[{point[[1]], point[[2]], f[point[[1]], point[[2]]]}]
    , Arrow[{ per = 0;
      {point[[1]] - per D[f[x, y], x] /. {x -> point[[1]], 
         y -> point[[2]]},
       point[[2]] -  per D[f[x, y], y] /. {x -> point[[1]], 
         y -> point[[2]]},
       f[point[[1]], point[[2]]] + per},
      {point[[1]] + D[f[x, y], x] /. {x -> point[[1]], 
         y -> point[[2]]},
       point[[2]] +  D[f[x, y], y] /. {x -> point[[1]], 
         y -> point[[2]]},
       f[point[[1]], point[[2]]] - 1}
      }]
    }]
  ,
  Graphics3D[{
    Texture[
     ContourPlot[f[x, y], {x, xmin, xmax}, {y, ymin, ymax}, 
      Axes -> False, PlotRangePadding -> 0, Frame -> False,
      Epilog -> {Red, Arrowheads[Large], PointSize[Large], 
        Thickness[Large], Point[{point[[1]], point[[2]]}]
        , Arrow[{{point[[1]], point[[2]]},
          {point[[1]] + D[f[x, y], x] /. {x -> point[[1]], 
             y -> point[[2]]},
           
           point[[2]] +  D[f[x, y], y] /. {x -> point[[1]], 
             y -> point[[2]]}}
          }]
        }
      ]]
    , EdgeForm[], 
    Polygon[{{xmin, ymin, zlevel}, {xmax, ymin, zlevel}, {xmax, ymax, 
       zlevel}, {xmin, ymax, zlevel}},
     VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}}]},
   Lighting -> "Neutral"
   
   ]
  , PlotRange -> All, BoxRatios -> {1, 1, 1}, FaceGrids -> {Back, Left}
  ]
 ,
 Row[{Style["f", Italic], "(", Style["x", Italic], ",", 
   Style["y", Italic], ") ="}],
 {{f, Function[{x1, y1}, 2 - (x1/2)^2 - (y1/2)^2], ""}, {
   Function[{x1, y1}, 2 - (x1/2)^2 - (y1/2)^2] -> 
    TraditionalForm[2 - (x/2)^2 - (y/2)^2]
   , Function[{x1, y1}, 1.5 Sin[x1 y1]] -> TraditionalForm[Sin[x y]]
   , Function[{x1, y1}, 1/3 x1 E^(y1/4) + 1/2 Cos[x1 y1]] -> 
    TraditionalForm[1/3 x E^(y/4) + 1/2 Cos[x y]]
   , Function[{x1, y1}, x1*Cos[y1]] -> TraditionalForm[x*Cos[y]]
   , Function[{x1, y1}, (x1^2 - y1^2)/2] -> 
    TraditionalForm[(x^2 - y^2)/2]
   },
  PopupMenu},
 Delimiter,
 Row[{"(", Subscript[Style["x", Italic], 0], ", ", 
   Subscript[Style["y", Italic], 0], ") = "}],
 {{point, {Mean[{.5 xmin, xmax}], Mean[{.6 ymin, ymax}]}, ""},
   {xmin, ymin}, {xmax, ymax}, ControlType -> Slider2D},
 TrackedSymbols :> {f, point},
 AutorunSequencing -> {1, 2},
 ControlPlacement -> Left
 ]