This is an interactive demonstration of what the directional derivative means. Suppose the wire mesh represents the surface of a mountain and a nearby valley. The green dot represents a mountain climber's position. As you change the bearing by sliding the blue dot around, you get an interactive look at the "cross section" of the surface through that point in that direction. Now imagine fitting a tangent line to the curve representing the cross section. That's the red arrow, where the direction of the arrow shows you which direction the climber is facing.

If the arrow is pointing up, then the directional derivative in that direction is positive. If the arrow is pointing down, then the directional derivative is negative. An arrow which is pointing just ever so slightly up would indicate a small (but positive) value for the directional derivative, say 0.01. If the arrow is tilted more upward, the derivative has a much higher positive value.

As you can see, it's positive as you move from the east, through the north, to the west. From the west, to the south, and back to the east, it's negative.