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 $P(5, 2, 5)$ in the direction of the vector $\mathbf{v} = \mathbf{i} + \mathbf{j} - \mathbf{k}$.
(b) In which direction does $V$ change most rapidly at $P$?
(c) What is the maximum rate of change at $P$?