Explanation:
To determine whether a system is linear or not, we need to check if it satisfies the properties of linearity. The properties of linearity are as follows:

1. Additivity: If x1(t) produces y1(t) and x2(t) produces y2(t), then x1(t) + x2(t) produces y1(t) + y2(t).
2. Homogeneity: If x(t) produces y(t), then ax(t) produces ay(t), where 'a' is a constant.

Now, let's analyze each system individually to determine if they are linear or not.

System P:
The output of System P is given by y(t) = dx(t)/dt.

- Additivity: Let's assume that x1(t) and x2(t) produce y1(t) and y2(t) respectively. Then, x1(t) + x2(t) produces y(t) = d(x1(t) + x2(t))/dt = d(x1(t))/dt + d(x2(t))/dt = y1(t) + y2(t). Therefore, System P satisfies additivity.
- Homogeneity: Let's assume that x(t) produces y(t). Then, ax(t) produces y(t) = d(ax(t))/dt = a(dx(t)/dt) = ay(t). Therefore, System P satisfies homogeneity.

Since System P satisfies both additivity and homogeneity, it is a linear system.

System Q:
The output of System Q is given by y(t) = ex(t).

- Additivity: Let's assume that x1(t) and x2(t) produce y1(t) and y2(t) respectively. Then, x1(t) + x2(t) produces y(t) = ex1(t) + ex2(t). However, this cannot be expressed as y1(t) + y2(t), as it violates additivity. Therefore, System Q does not satisfy additivity.
- Homogeneity: Let's assume that x(t) produces y(t). Then, ax(t) produces y(t) = e(ax(t)) = e^(ax(t)). This cannot be expressed as ay(t), as it violates homogeneity. Therefore, System Q does not satisfy homogeneity.

Since System Q does not satisfy both additivity and homogeneity, it is not a linear system.

System R:
The output of System R is given by y(t) = 3x(t).

- Additivity: Let's assume that x1(t) and x2(t) produce y1(t) and y2(t) respectively. Then, x1(t) + x2(t) produces y(t) = 3(x1(t) + x2(t)) = 3x1(t) + 3x2(t) = y1(t) + y2(t). Therefore, System R satisfies additivity.
- Homogeneity: Let's assume that x(t) produces y(t). Then, ax(t) produces y(t) = 3(ax(t)) = 3ax(t) = ay(t). Therefore, System R satisfies homogeneity.

Since System R satisfies both additivity and homogeneity, it is a linear system.

System S:
The output of System S is given by

System P