The rate of doing work by a force acting on a particle is given by the dot product of the force and the velocity of the particle. In this case, the force is given by 2x, and the velocity is given by an expression which we need to determine.

To find the velocity expression, we can use the fact that the rate of doing work is equal to the derivative of the work done with respect to time. Since the force is acting along the x-axis, the work done is given by the integral of the force with respect to displacement. Therefore:

Work done = ∫(2x) dx

Integrating the above expression, we get:

Work done = x^2 + C

where C is the constant of integration.

Since the rate of doing work is equal to the derivative of the work done with respect to time, we can equate the above expression to the derivative of the work done with respect to time:

x^2 + C = d/dt (x^2 + C)

Differentiating both sides with respect to time, we get:

0 = 2x(dx/dt)

Simplifying the above equation, we find:

dx/dt = 0

This implies that the velocity of the particle is constant and equal to zero.

Therefore, the velocity of the particle is given by expression:

dx/dt = 0

So, the correct answer is not provided in the options A, B, C, and D.

To summarize:
- The rate of doing work by a force is given by the dot product of the force and the velocity of the particle.
- The force acting on the particle is given by 2x.
- The velocity of the particle expression is determined by the derivative of the work done with respect to time.
- Integrating the force expression gives the work done expression as x^2 + C.
- Equating the work done expression to its derivative with respect to time gives a constant velocity of zero.
- None of the options A, B, C, and D provide the correct expression for the velocity of the particle.