Directional derivatives and gradients

[Imagine]

If f(x, y, z)=3xy+z^2 and u is the unit vector in the direction of (0, -2, 2), then the directional derivative at the point (−4−1−4) in the direction of u (Duf(−4,−1, −4)) is?

Duf(-4, -1, -4) = gradient of f at (-4, -1, -4) * (0, -2, 2)
gradient of f = (3y, 3x, 2z)
Plug in (-4, -1, -4) and get:
(-3, -12, -8)
Duf(-4, -1, -4) = (-3, -12, -8) * (0, -2, 2) =
0 + 24 - 16 = 8

This answer is wrong, and I have no idea what went wrong.

[Kalovale]

Directional derivative denotes the rate of change of the function value when you take one unit-length step along the "directional" vector in parameter space.

They gave you a vector u in the direction of the vector <0, -2, 2>, u itself is not <0, -2, 2>.

In short: