You titled this "directional derivatives and the gradient vector" so I would guess that somewhere in the chapter in which this problem appears you are told that if is a unit vector, then the derivative of f in the direction of is equal to the dot product .

What is here?

What is a unit vector pointing "due south", that is, in the negative y direction?a. if you are walk due south, will you start to ascend or descend? At what rate?

what is a vector pointing "northwest", that is, at 45 degrees between the negative x-axis and the positive y-direction?b. if you are walk northwest, will you start to ascend or descend? At what rate?

The gradient of z, always points in the direction of "fastest ascent". What is the length of the gradient vector? What angle does it make with the x-axis?c. in which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?

[quote]Any help is greatly appreaciated!!!!!!