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Newton's Second Law is fundamental in understanding motion. It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. This can be represented as \(\text{F = ma}\). Here, \(\text{F}\) refers to force, \(\text{m}\) to mass, and \(\text{a}\) to acceleration. This law helps us understand how the velocity of an object changes when it is subjected to an external force. For instance, when a car accelerates, the engine produces a force. This force changes the car's speed according to its mass. The bigger the car's mass, the more force you need to change its velocity. This principle is also used in the problem where the force is given and the mass of the particle is known. By rearranging Newton’s second law formula, we can solve for unknowns such as acceleration or the constant 'c' in this scenario.

Differentiation is a powerful tool in physics used to determine rates of change. When we differentiate a position function with respect to time, we get the velocity function. Velocity tells us how fast an object changes its position. For example, for the given motion equation \(\text{x(t) = 3.0 m + (4.0 m/s) t + c t^2 - (2.0 m/s^3) t^3}\), differentiating with respect to time gives us the velocity function: \(\text{v(t) = 4.0 m/s + 2c t - 6.0 (m/s^3) t^2}\). Similarly, differentiating the velocity function gives us the acceleration function. Acceleration is the rate at which velocity changes over time. Therefore, differentiating not only helps us find velocity from position but also acceleration from velocity.

Acceleration can be calculated by differentiating the velocity function with respect to time. In this exercise, we initially determine the equations of motion and velocity. Then we differentiate the velocity function to obtain the acceleration function: \(\text{a(t) = 2c - 12.0 m/s^3 t}\). Next, we substitute the given time into this function. For example, at \(\text{t = 3.0 s}\), the acceleration is \(\text{a(3.0 s) = 2c - 36.0 m/s^2}\). To determine the value of 'c', we use Newton's second law, which relates force and acceleration. Given the force and mass of the particle, we solve for 'c' and find its value to be \(\text{9 m/s^2}\). Understanding how to derive and use these equations is crucial for solving many physics problems related to motion.