Question

Kepler's third law derivation
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Answer 1

Kepler all plants move about the sun in elliptical orbit, having the sun as one of the foci

Answer 2

Orbital speed = √ GM/r
for any celestial or man made body revolving in a gravitational field of a desired celestial body.
So, M = mass of the body around which the body is revolving and r = distance between the centre of gravities of the body revolving and the body around which the other body is bring revolved.

Time period = Total revolution distance / Speed
= 2πr / √GM/r
Therefore,
T = 2π * r^3/2 / √GM
Hence squaring both sides,
T^2 = 4π^2 / GM. * r^3
Hence, proved that T^2 is proportional to r^3
That is the third law given by Kepler.
Note:- The orbits in orbital mechanics for simpler derivations and research , the elliptical orbits are nearly considered to be almost circular.
If you want to know how orbital speed = √GM/r then do tell me down in the comments

Hope this helps you !