Final answer:
To find the directional derivative of the function f = √xyz at the point (3, 2, 6) in the direction of vector v = (-1, -2, 2), we can use the gradient and dot product formulas. By calculating the gradient of f and evaluating the dot product with v, we can find the directional derivative.
Step-by-step explanation:
To find the directional derivative of the function f = √(xyz) at the point (3, 2, 6) in the direction of vector v = (-1, -2, 2), we can use the formula:
Dvf = ∇f · v
where ∇f is the gradient of f and · represents the dot product.
The gradient of f is given by:
∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
Plugging in the values of x=3, y=2, and z=6, we can find the gradient:
∇f = (√(yz), √(xz), √(xy)) = (√(2×6), √(3×6), √(3×2)) = (2√3, 2√2, 2√2)
Finally, we can calculate the directional derivative:
Dvf = (∇f · v) / |v|
Substituting the values, we get:
Dvf = (2√3 × -1 + 2√2 × -2 + 2√2 × 2) / |-1, -2, 2|
Dvf = (-2√3 - 4√2 + 4√2) / 3√3
Simplifying further, we get:
Dvf = (-2√3) / 3√3
Therefore, the directional derivative of f=√xyz at (3,2,6) in the direction v=(-1,-2,2) is (-2√3) / 3√3.