Plot3D [
      (Sqrt[((-(((x))^2))/1) - ((((y))^2)/1) + 
          (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .1]) , {x, -1, 1}, 
       {y,  -1,  1},  PlotPoints  -> 50,  Mesh -> False,  
      Boxed -> False]

D[(Sqrt[((-(((x))^2))/1) - ((((y))^2)/1) + 
          (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .125]), x]

D[(Sqrt[((-(((x))^2))/1) - ((((y))^2)/1) + 
          (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .125]), y]

D[(Sqrt[((-(((x))^2))/1) - ((((y))^2)/1) + 
          (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .125]), z]

z =.

Plot3D [((-2 x + (x/(Sqrt[(x^2) + (y^2)])))/(2 (Sqrt[-0.125`-(x^2)-(y^2)+(Sqrt[(x^2) + (y^2)])]))),{x,  -1,  1}, {y,  -1,  1} , PlotPoints  -> 50, Mesh -> False, Boxed -> False]

Plot3D [((-2 y + (y/(Sqrt[(x^2) + (y^2)])))/(2 (Sqrt[-0.125`-(x^2)-(y^2)+(Sqrt[(x^2) + (y^2)])]))),{x,  -1,  1}, {y,  -1,  1} , PlotPoints  -> 50, Mesh -> False, Boxed -> False]

D[D[
        (Sqrt[((-(((x))^2))/1) - ((((y))^2)/1) + 
            (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .125]), x], x]

D[D[
        (Sqrt[((-(((x))^2))/1) - ((((y))^2)/1) + 
            (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .125]), y], y]

D[D[
        (Sqrt[((-(((x))^2))/1) - ((((y))^2)/1) + 
            (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .125]), y], x]

Plot3D [(((-2 x + (x/(Sqrt[(x^2) + (y^2)]))) 
              (-2 y + (y/(Sqrt[(x^2) + (y^2)]))))/(4 (((-0.125`-(x^2)-(y^2)+(Sqrt[(x^2) + (y^2)])))^(3/2))))-((x y)/(2 ((((x^2) + (y^2)))^(3/2)) (Sqrt[-0.125`-(x^2)-(y^2)+(Sqrt[(x^2) + (y^2)])]))), {x, -1,  1}, {y,  -1,  1} , PlotPoints  -> 100, Mesh -> False, Boxed -> False]

Plot3D [-(((((-2 x + (x/(Sqrt[(x^2) + (y^2)]))))^2))/(4 (((-0.125`-(x^2)-(y^2)+(Sqrt[(x^2) + (y^2)])))^(3/2))))+((-2 - ((x^2)/((((x^2) + (y^2)))^(3/2))) + 
              (1/(Sqrt[(x^2) + (y^2)])))/(2 (Sqrt[-0.125`-(x^2)-(y^2)+(Sqrt[(x^2) + (y^2)])]))), {x, -1,  1}, {y,  -1,  1} , PlotPoints  -> 100, Mesh -> False, Boxed -> False]

Plot3D [-(((((-2 y + (y/(Sqrt[(x^2) + (y^2)]))))^2))/(4 (((-0.125`-(x^2)-(y^2)+(Sqrt[(x^2) + (y^2)])))^(3/2))))+((-2 - ((y^2)/((((x^2) + (y^2)))^(3/2))) + 
              (1/(Sqrt[(x^2) + (y^2)])))/(2 (Sqrt[-0.125`-(x^2)-(y^2)+(Sqrt[(x^2) + (y^2)])]))), {x, -1,  1}, {y,  -1,  1} , PlotPoints  -> 100, Mesh -> False, Boxed -> False]

Plot3D [
      (((((-(((x))^32))/1) - ((((y))^32)/1) + 
            (Sqrt[((((x))^32)/1) + ((((y))^32)/1)]) -  .1))^((1/32))) , {x, 
        -1.3, 1.3},  {y,  -1.3,  1.3},  PlotPoints  -> 200,  
      Mesh -> False,  Boxed -> False,  ImageSize -> 800]

Plot3D [
      (((((-(((x))^((1/2))))/1) - ((((y))^((1/2)))/1) + 
            (Sqrt[((((x))^((1/2)))/1) + ((((y))^((1/2)))/1)]) -  .1))^2) , {
        x, -1, 1},  {y,  -1,  1},  PlotPoints  -> 50,  
      Mesh -> False,  Boxed -> False]

Plot3D [
      (((((-(((x))^2))/1) - ((((y))^2)/1) + 
            (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .1))^((1/64))) , {x, 
        -1.3, 1.3},  {y,  -1.3,  1.3},  PlotPoints  -> 200,  
      Mesh -> False,  Boxed -> False,  ImageSize -> 800]

Plot3D [
      (((((-(((x))^2))/1) - ((((y))^2)/1) + 
            (Sqrt[((((x))^2)/1) + ((((y))^2)/1)]) -  .1))^((1/256))) , {x, 
        -1.3, 1.3},  {y,  -1.3,  1.3},  PlotPoints  -> 200,  
      Mesh -> False,  Boxed -> False,  ImageSize -> 800]