Write down the mathematical derivation of …

Let the work done on the object against gravity = W
Work done, W = force × displacement
Work done, W  = mg × h
Work done, W = mgh
Since workdone on the object is equal to mgh, an energy equal  to mgh units is gained by the object . This is the potential energy (Ep) of the object.
 E_p = mgh
Derivation for the equation of Kinetic Energy:
The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, S is 
 v² - u² = 2aS
This gives  
 S = (v ² - u ²)/  2a
We know F = ma. Thus using above equations, we can write the workdone by the force, F as

  W = ma ×{ v ² - u ²/ 2a}
   or
 W = 1 /2 m( v ² - u ² )
If object is starting from its stationary position, that is, u = 0, then
W = 1/2 m v 2
It is clear that the work done is equal to the change in the kinetic energy of an object.
If u = 0, the work done will be  W = 1/2 m v ² .

Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is E_k =  ½  mv²