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 migh slope steeply up in one direction, be relative 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 maximal slope.
Graphics on the left is top view of the 3D scene on the right. Specify the point A on paraboloid a with the coordinates: A1= (x0,y0). Direction is given by angle a. Length of the vector A1D is equal to the directional derivative dd.