I guess I am a little confused on how to differentiate the difference between directional derivative vs. total differential. The total differential is a place holder for the partial derivatives so if you tile a vector to them then there is your directional derivative and gradient. And if we sketch a level curve at the value then isn't the gradient also a directional derivative. The only difference I see is that there are a lot more directional derivatives because they are constructed with unit vectors so you can make them anything you want but the function w at some particular point only has ONE possible solution to where and how far it points and that is certainly a directional derivative. Furthermore that very directional derivative must be a gradient to the level curve at the point. or am I all wet on the intuition. Thank you much
The gradient of a function, at a given point, is the directional derivative in the direction in which the directional derivative has largest norm. And that is the direction perpendicular to a level curve at that point.