del=(1/(Sqrt[(x^2)+(y^2)+(z^2)]))-((1/(Sqrt[(((x+1))^2)+(y^2)+(z^2)]))+(1/(Sqrt[(((x-1))^2)+(y^2)+(z^2)]))+(1/(Sqrt[(x^2)+(((y+1))^2)+(z^2)]))+(1/(Sqrt[(x^2)+(((y-1))^2)+(z^2)]))+(1/(Sqrt[(x^2)+(y^2)+(((z+1))^2)]))+(1/(Sqrt[(x^2)+(y^2)+(((z-1))^2)])))/6

Simplify[del]

del /. Thread[{x,y,z}->RandomReal[{-100,100},3]]

Plot3D[del/.z->0, {x,-10,10},{y,-10,10}]

Δ[f_][x_,y_,z_] := f[x,y,z]-(f[x+1,y,z]+f[x-1,y,z]+f[x,y+1,z]+f[x,y-1,z]+f[x,y,z+1]+f[x,y,z-1])/6

f[x_,y_,z_]:=1/(1+Total[Abs/@{x,y,z}]);
Table[Δ[f][x,y,1], {x,0,10},{y,0,10}]//MatrixForm