Suppose that over a certain region of space the electrical potential V is given by the following equation.
$V(x, y, z) = 3x^2 - 3xy + xyz$
(a) Find the rate of change of the potential at $P(4, 4, 5)$ in the direction of the vector $v = i + j - k$.
(b) In which direction does V change most rapidly at P?
(c) What is the maximum rate of change at P?