Directional Derivative:
The directional derivative of a function f(x, y, z) in the direction of a vector v = (a, b, c) is denoted as Dv(f) and represents the rate at which the function changes along the direction of the vector.

Given Function:
The given function is f = x^2 - y^2 + 2z^2.

Direction:
The direction in which we need to find the directional derivative is along the x-axis. This means the vector v will have the form v = (a, 0, 0), where a is a constant.

Finding the Directional Derivative:
To find the directional derivative, we will use the formula:

Dv(f) = ∇f · v

where ∇f represents the gradient of the function f and · denotes the dot product.

Gradient of the Function:
The gradient of a function f(x, y, z) is a vector that points in the direction of the steepest increase of the function at a given point. It is denoted as ∇f and is given by:

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Here, ∂f/∂x represents the partial derivative of f with respect to x, ∂f/∂y represents the partial derivative of f with respect to y, and ∂f/∂z represents the partial derivative of f with respect to z.

Partial Derivatives of the Given Function:
To find the partial derivatives of the given function, we differentiate f with respect to each variable separately.

∂f/∂x = 2x
∂f/∂y = -2y
∂f/∂z = 4z

Gradient of the Given Function:
Using the partial derivatives, we can find the gradient of the given function.

∇f = (2x, -2y, 4z)

Directional Derivative along the x-axis:
To find the directional derivative along the x-axis, we substitute the direction vector v = (a, 0, 0) into the gradient formula.

Dv(f) = ∇f · v
= (2x, -2y, 4z) · (a, 0, 0)
= 2ax

Substituting the Point P(1, 2, 3):
To find the directional derivative at point P(1, 2, 3), we substitute the values of x, y, and z into the expression obtained above.

Dv(f) = 2ax
= 2(1)(1)
= 2

Therefore, the directional derivative of f = x^2 - y^2 + 2z^2 at P(1, 2, 3) along the x-axis is 2 (option D).

Directional derivative of f = x² - y² + 2z² at P(1,2,3) along x-axis i.e., in the direction of is