Directional derivative
Partial derivative with respect to x gives the rate of change of f in the x direction and the partial derivative with respect to y gives the rate of change of f in the y direction.
The rate of change of a function of several variables in the direction u is called the directional derivative in the direction u. The directional derivative is the dot product of the gradient and the unit vector u, |u|=1.
z'u(A) = ∇z(A) . u
Note that if u is a unit vector in the x direction u = (1,0), then the directional derivative is simply the partial derivative with respect to x. For a general direction, the directional derivative is a combination of the partial derivatives. Problem: Cut the surface by vertical plane passing through the point A with the direction given by direction. Determine the slope of the intersection curve.