I want step by step derivation of least count.

Least count is the minimum measurement that can be made by a measuring device. 

 Least count of a vernier callipers = minimum measurement on main scale / total number of divisions on vernier scale. 

Let’s derive this:

Where S is the value of 1 Main scale division and N is the number of equal divisions on the vernier scale.

 

Least count of a screw gauge = minimum measurement on main scale / total number of divisions on

 

Smaller is the magnitude of least count of a measuring instrument, more precise the measuring.