Directional derivative and gradient.
Let the function a(x,y) be the height of a mountain range. If you stand at point A, the slope of the ground in front of you will depend on the direction you are facing. It might slope steeply up in one direction, be relatively flat in another direction, and slope steeply down in yet another direction.
The partial derivatives give the slopes in the positive x and y directions. Directional derivatives generalize the partial derivatives to calculate the slope in any direction.
Gradient points in the direction of the maximal slope.
Graphics on the left is a top view of the 3D scene on the right.
Specify point A on paraboloid a with the coordinates: A1= (x0,y0). Direction is given by angle α.
The length of the vector A1D is equal to the directional derivative dd.