Final Answer:
The directional derivative of the function f(x, y, z) = xyz in the direction of u = (6, 2, -9) is -11x² + 6yz - 11xz.
Step-by-step explanation:
The directional derivative of a scalar field f at a point p in the direction of a vector u is given by the formula:
∇f · u = Duf = ∂f/∂x * ux + ∂f/∂y * uy + ∂f/∂z * uz
where ∇f is the gradient of f, ux, uy, and uz are the components of u.
In this case, we have:
f(x, y, z) = xyz
∇f = (yz, xz, xy)
u = (6, 2, -9)
Therefore, the directional derivative Duf is calculated as follows:
Duf = ∇f · u = (yz, xz, xy) · (6, 2, -9)
= yz * 6 + xz * 2 + xy * (-9)
= -11x² + 6yz - 11xz
Therefore, the directional derivative of the function f(x, y, z) = xyz in the direction of u = (6, 2, -9) is -11x² + 6yz - 11xz.