Final answer:
To find the directional derivative of f(x, y) = √(xy) at point P(8,8) in the direction from P to Q(11, 4), we need to find the gradient vector and then take the dot product with the unit vector in the direction from P to Q.
Step-by-step explanation:
To find the directional derivative of f(x, y) = √(xy) at point P(8,8) in the direction from P to Q(11, 4), we need to find the gradient vector and then take the dot product with the unit vector in the direction from P to Q.
The gradient of f(x, y) is given by ∇f = (∂f/∂x, ∂f/∂y) = (√(y/2x), √(x/2y)). Evaluating this at P(8,8) gives ∇f(P) = (√1, √1) = (1, 1).
The direction vector from P to Q is given by Q - P = (11-8, 4-8) = (3, -4). To obtain the unit vector in this direction, we divide the direction vector by its magnitude: (3, -4)/√(3^2 + (-4)^2) = (3/5, -4/5).
Finally, taking the dot product of ∇f(P) and the unit vector, we get the directional derivative as (1, 1) · (3/5, -4/5) = 3/5 - 4/5 = -1/5. Therefore, the directional derivative of f(x, y) at P(8,8) in the direction from P to Q is -1/5.