Section 14.6: Directional Derivatives & Gradient Vector

Suppose that over a certain region of space the electrical potential V is given by \(V(x, y, z) = 5x^2 - 3xy + xyz\). (a) Find the rate of change of the potential at \(P(3, 4, 5)\) 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?