Final answer:
The derivative of the function f(x,y,z) at the point (-1,1,3) in the direction of the vector A = 2i + 9j - 6k is calculated by first finding the gradient of f at that point and then computing the dot product of that gradient with the unit direction vector of A.
Step-by-step explanation:
The student is asking to find the derivative of the function f(x,y,z)=xy+yz+zx at the point P0 (−1, 1, 3) in the direction of the vector A = 2i + 9j − 6k. To solve this, we first need to find the gradient of the function f, and then calculate the directional derivative in the direction of A.
First, we calculate the partial derivatives of f with respect to x, y, and z:
- ∂f/∂x = y + z
- ∂f/∂y = x + z
- ∂f/∂z = x + y
The gradient of f at point P0 is then ∇f|P0 = (4, 2, 0).
We then normalize the vector A to find the unit vector in the direction of A. The unit vector uA is (2/11)i + (9/11)j − (6/11)k.
Finally, the directional derivative of f at P0 in the direction of A is given by ∇f|P0 · uA = (4)(2/11) + (2)(9/11) + (0)(-6/11).