Final answer:
The directional derivative of the function at the origin in the direction of the vector (6, 3, -1) is found by computing the gradient of the function at the origin, normalizing the direction vector, and taking the dot product of the two.
Step-by-step explanation:
To find the directional derivative of a function f(x, y, z) at a point, one must first calculate the gradient of the function at that point. Next, the directional derivative in the direction of a given vector v is the dot product of the gradient vector and the unit vector in the direction of v. For the given function f(x, y, z) = xey yez zex at the point (0, 0, 0), and the vector v = (6, 3, -1), we first normalize v to get the unit vector in its direction. After computing the gradient of f at the origin, we take the dot product with the unit vector to obtain the directional derivative.
The normalized vector v is obtained by dividing v by its magnitude. Then, the gradient of f at the origin (0, 0, 0) is the vector of partial derivatives of f. We compute each partial derivative and evaluate at the origin. Finally, we calculate the dot product of the gradient vector and the normalized direction vector to find the main answer.