Gauss's theorem, also known as Gauss's law, is a fundamental principle in electromagnetism that relates the electric flux passing through a closed surface to the total charge enclosed by that surface. It is an important tool for solving problems in electrostatics, and it has several key properties that make it a useful and accurate theorem.

a) Gauss's theorem can be derived by using Coulomb's law. Coulomb's law describes the force between two charged particles, and it states that the force is proportional to the product of the charges and inversely proportional to the square of the distance between them. By applying Coulomb's law to a small test charge placed at various points within a closed surface, we can derive Gauss's theorem. Therefore, option 'a' is correct.

b) Gauss's theorem is valid for conservative fields and obeys the inverse square root law. A conservative field is one in which the work done in moving a particle between two points is independent of the path taken. The electric field generated by a static charge distribution is a conservative field, and Gauss's theorem applies to such fields. Additionally, the electric field follows the inverse square root law, meaning that its strength decreases with the square of the distance from the source charge. Therefore, option 'b' is correct.

c) Gauss's theorem is not applicable in gravitation. While Gauss's theorem is a fundamental principle in electromagnetism, it does not directly apply to gravitational fields. The gravitational field generated by a mass distribution does not obey Gauss's theorem, and different principles, such as Gauss's law for gravity, need to be applied. Therefore, option 'c' is correct.

d) As discussed above, option 'd' is incorrect because Gauss's theorem can indeed be derived using Coulomb's law and it is valid for conservative fields that obey the inverse square root law.