Suppose $f(x,y)$ is a differentiable function and $v = (a, b)$ is a vector.
If $(x_0,y_0) \in D_f$ and $\frac{\partial f}{\partial v}(x_0,y_0) = 0$.
What is the meaning of this? Along the direction of the vector $v$ the graph does not change the $z$ coordinate?
It means that if you start at $(x_0,y_0)$, and take a small step in the direction $v$, the value of $f$ doesn't change (very much).
We could make that a bit more precise by saying that when you take a small step in the direction $v$, the change in $f$ is small when compared with the distance that you moved.
(We could make this more precise but then we'd be stating the formal definition of the directional derivative.)
