Final answer:
The directional derivative of f(x, y, z) = z³ - x²y at the point (-5, 1, -4) in the direction of the vector v = <2, 4, -2> is -176.
Step-by-step explanation:
To find the directional derivative of the function f(x, y, z) = z³ - x²y at the point (-5, 1, -4) in the direction of the vector v = <2, 4, -2>, we can use the formula:
Dvf(a, b, c) = ∇f(a, b, c) · v
where ∇f(a, b, c) is the gradient of the function at the point (a, b, c). In this case, the gradient is:
∇f(x, y, z) = (-2xy, -x², 3z²)
Plugging in the values (-5, 1, -4), we get:
∇f(-5, 1, -4) = (10, -25, 48)
Finally, we can calculate the directional derivative:
Dvf(-5, 1, -4) = ∇f(-5, 1, -4) · v = (10, -25, 48) · (2, 4, -2) = 20 - 100 - 96 = -176