Derivation of transcondutance gm using Taylor expansion
As stated in the introduction, all other voltages are held constant:


  Derivation: consider for example the relationship of the increment in drain
  current due to an increment in gate-source voltage when the MOSFET is
  saturated-- with all other voltages held constant.


An for this reason only the change of the gate-source voltage is considered.

Of course to extend this model a similar derivation will be done for vds and vbs resulting in gds and gmbs.