Could be a bit hard, because Lambert W function is not an elementary function.
If you would like to deriv similar formula using square law approximation, check Gray, Hurst, Meyer book "Analysis and Design...", chapter related to diff pairs.
Edit: If you want the proof for MOSFETs, it can be found in Razavi's 'Fundamental of Microelectronics'. I have extracted the most important parts of the expressions..
But that is just the small signal current. All of these expressions were derived NOT assuming small signal models (and thus all the complicated expressions)... And thus, I assumed the DC component of the tail current is Io.
EDIT 1: (Some more comments)
I cannot see it being anything other than a DC component anyway.