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A similar derivation is given in the first edition of Kittel’s Thermal Physics, as well as in the Quantum Mechanics book by Griffiths IIRC.

Feynman has a similar dicussion starting from the P=nkT equation: http://www.feynmanlectures.caltech.edu/I_40.html

Does Kittel define Shannon entropy?

I'd have to grab the first edition copy from the library but I think not, he does use Lagrange multipliers though.