Manipulate[Module[{x, y},
  ContourPlot[Exp[-x^2 - y^2] + x y, {x, -1, 1}, {y, -1, 1},
   Contours -> 20,
   Epilog ->
    Dynamic[{Arrow[{pt,
        pt + {y - 2 E^(-x^2 - y^2) x, x - 2 E^(-x^2 - y^2) y} /. {x ->
            pt[[1]], y -> pt[[2]]}}]}]
   ]],
 {{pt, {.5, .5}}, Locator,
  Appearance -> Graphics[{Red, Disk[]}, ImageSize -> 5]}]

Clear[x, y, f];
f = E^(-x^2 - y^2) + x y;
Grad[f, {x, y}];

Manipulate[Module[{x, y},
  ContourPlot[f, {x, -1, 1}, {y, -1, 1},
   Contours -> 20,
   Epilog ->
    Dynamic[{Arrow[{pt,
        pt + Grad[f, {x, y}] /. {x -> pt[[1]], y -> pt[[2]]}}]}]
   ]],
 {{pt, {.5, .5}}, Locator,
  Appearance -> Graphics[{Red, Disk[]}, ImageSize -> 5]}]

Manipulate[
 ContourPlot[f, {x, -1, 1}, {y, -1, 1},
  Contours -> 20,
  Epilog ->
   Dynamic[{Arrow[{pt,
       pt + Grad[f, {x, y}] /. {x -> pt[[1]], y -> pt[[2]]}}]}]
  ],
 {{pt, {.5, .5}}, Locator,
  Appearance -> Graphics[{Red, Disk[]}, ImageSize -> 5]}]

x=12

Clear[x, y, f];
f = E^(-x^2 - y^2) + x y;
Grad[f, {x, y}];

DynamicModule[{x, y},
 Manipulate[
  ContourPlot[f, {x, -1, 1}, {y, -1, 1},
   Contours -> 20,
   Epilog ->
    Dynamic[{Arrow[{pt,
        pt + Grad[f, {x, y}] /. {x -> pt[[1]], y -> pt[[2]]}}]}]
   ],
  {{pt, {.5, .5}}, Locator,
   Appearance -> Graphics[{Red, Disk[]}, ImageSize -> 5]}]]