anyone konw what its meaning?

On that surface, calculus looks for the rate of change (or the slope). There is a
directional derivative, whatever the direction. In the 45" case we are inclined to divide Af by Ax, but we would be wrong.
Let me state the problem. We are given f(x, y) around a point P = (x,, yo). We are also given a direction u (a unit vector). There must be a natural definition of D,f-the derivative off in the direction u. To compute this slope at P, we need a formula.
Preferably the formula is based on df/dx and dfldy, which we already know.
Note that the 45" direction has u = i/[img]$[/img] + j/$. The square root of 2 is going to
enter the derivative. This shows that dividing Af by Ax is wrong. We should divide by the step length As.