How was the Cobb Douglas function derived?
What is the proof of this formula?

There is actually no proof for what the production function should be. There are infinite many possible production functions and to discover which one is the most appropriate we need to make some empirical observations. In different cases different production functions are appropriate. Cobb-Douglas is popular production function but I seen many others as well.
What you provide below in your update is not so much of a proof that production has to be Cobb-Douglas rather its a proof that if we make some specific assumptions about production (which although quite general might not always necessary hold actually) we get function that is Cobb-Douglass.

$ln(P)=α∗ln(cL)$
For example here, from where it comes the constant &quot;c&quot;?,

the $c$ is an integration constant. Whenever, you have indefinite integrals you have to add some constant $c$ to the solution because constants get eliminated during differentiation we can never know if there was or wasn't some constant previously so after integrating we always add $c$.
In this case when you integrate this separable differential equation the solution would actually look like:
$$\ln(P)= \alpha (\ln(L) + C) \implies\ln(P)= \alpha \ln(cL) | C= \ln(c) $$
(actually the variables should even be in absolute values - but since the function is only defined for non-negative values of $P,L$ and $K$ we can omit them).