**Newton's Second Law of Motion**

Newton's second law of motion states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass. Mathematically, it can be expressed as:

F = ma

Where:
F is the net force acting on the object,
m is the mass of the object, and
a is the acceleration produced.

**Deriving Newton's First Law of Motion**

Newton's first law of motion, also known as the law of inertia, states that an object at rest will stay at rest, and an object in motion will continue in motion with a constant velocity unless acted upon by an external force. We can derive this law from Newton's second law by considering the scenario where the net force acting on an object is zero (F = 0).

1. When F = 0, the equation F = ma becomes 0 = ma.
2. Since the mass (m) of the object is non-zero, the equation simplifies to 0 = a.
3. This means that when the net force is zero, the acceleration of the object is zero.
4. If the acceleration is zero, it implies that the object's velocity remains constant.
5. Therefore, an object at rest (zero velocity) will remain at rest, and an object in motion (non-zero velocity) will continue moving with a constant velocity in a straight line, in the absence of any external force.
6. This is consistent with Newton's first law of motion.

**Deriving Newton's Third Law of Motion**

Newton's third law of motion states that for every action, there is an equal and opposite reaction. To derive this law from Newton's second law, we need to consider a system of two interacting objects.

1. Let's consider two objects, A and B, with masses mA and mB, respectively, and let them exert forces on each other (FA and FB).
2. According to Newton's second law, the acceleration of object A is given by aA = FA / mA, and the acceleration of object B is given by aB = FB / mB.
3. Now, let's consider the net force acting on the system of both objects. The net force is the sum of the forces exerted by A and B, which can be expressed as FA + FB.
4. Using Newton's second law, we can write mA * aA + mB * aB = FA + FB.
5. Rearranging the equation, we get mA * aA - FA = FB - mB * aB.
6. Notice that -FA is the force exerted by object A on B, and FB is the force exerted by object B on A.
7. Therefore, the equation can be rewritten as mA * aA = -FAB and mB * aB = FBA.
8. This demonstrates that the forces between the two objects A and B are equal in magnitude but opposite in direction.
9. Hence, we have derived Newton's third law of motion, which states that for every action force (FAB) exerted by object A on B, there is an equal and opposite reaction force (FBA) exerted by object B on A.