The drain conductance of n-channel MOSFET in linear region is:

If VDS < VGS - VT

Then MOSFET is in the active region (linear region)

Drain current is given by

\({{I}_{D}}=\frac{1}{2}{{\mu }_{n}}{{C}_{ox}}\frac{W}{L}\left( 2\left( {{V}_{GS}}-{{V}_{T}} \right){{V}_{DS}}-V_{DS}^{2} \right)\)

If VDS ≥ VGS - VT

The mode of operation is saturation in the region.

\({{I}_{D\left( sat \right)}}={{\mu }_{n}}{{C}_{ox}}\frac{W}{2L}{{\left( {{V}_{GS}}-{{V}_{T}} \right)}^{2}}\)

For trans-conductance, we differentiate drain current w.r.t. VGS

In the active region (Linear region)

\({{g}_{m}}=~\frac{d{{I}_{D}}}{d{{V}_{GS}}}=\frac{W}{2L}{{\mu }_{n}}{{C}_{ox}}\left( 2{{V}_{DS}} \right)=\frac{W}{L}{{\mu }_{n}}{{C}_{ox}}{{V}_{DS}}\)

So, drain conductance of n-channel MOSFET in the linear region
= \(\frac{W}{L}{{\mu }_{n}}{{C}_{ox}}{{V}_{DS}}\)

Additional Information

In saturation region

\({{g}_{m}}=\frac{d{{I}_{{{D}_{sat}}}}}{d{{V}_{GS}}}=\frac{W}{2L}{{\mu }_{n}}{{C}_{ox}}2\left( {{V}_{GS}}-{{V}_{T}} \right)=\frac{W}{L}{{\mu }_{n}}{{C}_{ox}}\left( {{V}_{GS}}-{{V}_{T}} \right)\)