Hold one input still and let the other move. You have sliced a surface into an ordinary curve.
The x-partial differentiates the curve obtained by fixing y. The y-partial fixes x instead. The highlighted slices and tangent segments show these two independent rates at the selected point.
Zero partial derivatives do not by themselves imply an extremum.
Drag to rotate, or use the view sliders. The surface and level curve are sampled; all point values and derivatives use exact formulas.