Directional Derivative:
The directional derivative of a function f(x, y, z) in the direction of a vector v = (a, b, c) at a point (x0, y0, z0) is the rate at which the function changes in the direction of the vector v at the point (x0, y0, z0).

Given Function:
The function f(x, y, z) = x^2 * y^2 * z^2

Direction:
The direction is given by the line x/3 = y/4 = z/5.

Step 1: Find the Unit Vector:
To find the unit vector in the direction of the line, we normalize the direction vector (a, b, c).

Let's assume a = x/3, b = y/4, and c = z/5.

Then, the unit vector in the direction of the line is given by:
u = (a, b, c) / sqrt(a^2 + b^2 + c^2)

Step 2: Evaluate the Directional Derivative:
The directional derivative of f(x, y, z) in the direction of the unit vector u is given by:

Duf(x0, y0, z0) = ∇f(x0, y0, z0) · u

where ∇f(x0, y0, z0) is the gradient of f(x, y, z) evaluated at (x0, y0, z0).

The gradient of f(x, y, z) is given by:
∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Step 3: Calculate the Gradient:
To calculate the gradient, we need to find the partial derivatives of f(x, y, z) with respect to x, y, and z.

∂f/∂x = 2xy^2z^2
∂f/∂y = 2x^2yz^2
∂f/∂z = 2x^2y^2z

Step 4: Evaluate the Directional Derivative:
Now, we can evaluate the directional derivative of f(x, y, z) in the direction of u at the point (1, 2, 3).

Substituting the given values into the partial derivatives:
∂f/∂x = 2(1)(2^2)(3^2) = 72
∂f/∂y = 2(1^2)(2)(3^2) = 72
∂f/∂z = 2(1^2)(2^2)(3) = 24

Next, we calculate the unit vector u:
a = 1/3, b = 2/4 = 1/2, c = 3/5

The magnitude of the vector u is:
|u| = sqrt((1/3)^2 + (1/2)^2 + (3/5)^2) = sqrt(1/9 + 1/4 + 9/25) = sqrt(25/100 +