How to derive gm and ro in a small signal models in term of large model parameters K, VT and bias voltages VGS and VDS, given that in a bias region iD = K(VGS-VT)(VDS^2)

 

Sounds like homework. If so it should be in that section.

 

Take the partial derivative of the drain current with respect to the drain-source voltage and invert it to find the output resistance.
\( r_o = \left( \frac{\partial i_d}{\partial v_{ds} \right) ^{-1} \)

Take the partial derivative of the drain current with respect to the gate-source voltage to find the transconductance.
\( g_m = \frac{\partial i_d}{\partial v_{gs} \)

All partial derivatives should be evaluated at the operating point -- that is, you should plug in the dc voltage/current values to evaluate the output resistance and the transconductance.