I've had some trouble with the (visual) intuition behind the directional derivatives so I decided to take a step back and look up the visual intuition behind partial derivatives, which I think I do understand. See picture below
As I understood it, we basically have the purple paraboloid (?) which is a function of (x,y) and then we have the gray plane which is the plane in the direction of the x-axis. If you intersect the 2 planes I would say you get a parabola. If you have a certain point specified on the paraboloid, you can find its partial derivative in the direction of x.
The way I draw the connection to a directional derivative is just by saying that you can tilt the gray plane in any direction and find the derivative. Is this correct, and if not, what's wrong?