There are many ways to walk across a surface. The directional derivative gives the slope along one unit direction.
The dot product projects the gradient onto the direction of travel. Opposite directions give opposite rates. Perpendicular directions give zero first-order change. This formula applies to the differentiable surfaces offered here.
The direction vector must have length one to measure change per unit distance.
Drag to rotate, or use the view sliders. The surface and level curve are sampled; all point values and derivatives use exact formulas.