plotting - 3D Contour Plot parts of sphere with gradient less than 0, equal to 0 and more than 0

I want to plot only parts of the sphere of radius 1 where the gradient $\frac{\partial z}{\partial x}$ is < 0, = 0 and > 0.

The full sphere is given by:

ContourPlot3D[x^2 + y^2 + z^2 == 1, {x, -1, 1}, {y, -1, 1}, {z, -1, 1}]

I know how to plot parts of the sphere where $x^2 + y^2 + z^2 < 0.5 $ or something, but I can't figure out how to apply the condition to the gradient.

UPDATE: The positive and negative gradients turn out fine, but when I try to plot gradient = 0 it turns out funny.

I tried to find $\frac{\partial z}{\partial x} = 0$ for the equation but it didn't work: $$x^2 + y^2 + z^2 == xy $$

eq1 = x^2  y + y^2 + z^2 x == x y
ContourPlot3D[Evaluate[eq1], {x, -2, 1}, {y, -2, 2}, {z, -2, 2}]

deriv = Derivative[1, 0][z][x, y] /. 
    First[  Solve[    D[eq1 /. z ->   z[x, y], x],  
      Derivative[1, 0][z][x, y]  ]  ]  /. z[x, y] ->  z;

Positive1 = 
  ContourPlot3D[Evaluate[eq1], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   RegionFunction ->  Function[{x, y, z}, deriv > 0], Mesh -> False,  

   ContourStyle -> Blue, MaxRecursion -> 5];
Negative1 = 
  ContourPlot3D[Evaluate[eq1], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   RegionFunction ->  Function[{x, y, z}, deriv < 0], Mesh -> False, 
   ContourStyle -> Red, MaxRecursion -> 5];
zero1 = ContourPlot3D[
   Evaluate[eq1], {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
   RegionFunction ->  Function[{x, y, z}, deriv = 0], Mesh -> False, 
   ContourStyle -> Green, MaxRecursion -> 5];


Show[Positive1]
Show[Negative1]
show[zero1]