Answer :
Final answer:
The directional derivative of the function f(x, y, z) = z^3 - x^2 * y at the point (2, -5, -5) in the direction of the vector v = (-4, 1, 4) is -300.
Explanation:
The directional derivative of the function f(x, y, z) = z^3 - x^2 * y at the point (2, -5, -5) in the direction of the vector v = (-4, 1, 4) can be found using the formula:
D_v f(x, y, z) = gradient(f) * v
Plugging in the values, we can calculate:
D_v f(2, -5, -5) = (3z^2, -2xy, x^2) * (-4, 1, 4)
Simplifying, we get:
D_v f(2, -5, -5) = (-300, 20, -400)
So the directional derivative of the function in the given direction is -300.
