方向导数

The two coordinate directions were only the beginning

One dot product does all directions

The gradient points the way up

Three variables and the same story

For $f(x, y) = x^2 + 3y^2$ at $(1, 1)$, what is the directional derivative in the direction $\mathbf{u} = (3/5, 4/5)$?

For $f(x, y) = x^2 + 3y^2$ at $(1, 1)$, in which unit direction does the function increase fastest, and at what rate?

At a point where $\nabla f = (2, 6)$, in which directions is the directional derivative exactly $0$?

For $f(x, y, z) = x y z$ at the point $(1, 2, 3)$, give the first component of the gradient, as a plain number.

For $f(x, y) = x^2 + 3y^2$ at $(1, 1)$, give the rate of change along the direction of steepest descent, in the form -a*sqrt(b) with integers.