The directional derivative in the direction where (it is a unit vector) then
is the directional derivative.
First recall the geometric definition of the dot product
First note that this function takes on it maximum value when
Also since is a unit vector we get
Using the first observation this has a maximum when is parallel to so the gradient points in the direction of max rate of change.
For 2) since the function is constant on a level surface the directional derivative must be equal to zero. This would give