Intuitively, I think of the directional derivative as the slope of the graph in a particular direction. But this intuition seems flawed.
Consider the directional derivative of $f(x, y) = xy$ at the point $(0, 0)$ in the direction $(1, 1)$. The slope of the graph in this direction should be positive, but the directional derivative is $0$.
If the directional derivative doesn't represent the slope of the graph in some direction, what does it represent? How should I think of it intuitively?
Best Answer
It does represent the slope of the graph in that direction. The slope of $f(x,y)=xy$ at $(0,0)$ is $0$ in all directions, since there is a saddle point at $(0,0)$, that is, there are two perpendicular directions, at one of them the function has a local minimum, and at the other the function has a local maximum.
This shouldn't be surprising: it is similar to how in single variable calculus a function with a local extrema, say $f(x)=x^2$, or even a function with an inflection point, say $f(x)=x^3$, has derivative $0$ at that point.
The best linear approximation to $f(x)=x^2$ at $x=0$ is the line $y=0$, and in the same way, the best linear approximation to $f(x,y)=xy$ at $(0,0)$ is the plane $z=0$, and so in any direction the slope is $0$.