The transconductance of n-channel MOSFET in linear region is:

For a MOSFET in saturation, the current is given by:

\({I_{D\left( {sat} \right)}} = \frac{{W{μ _x}{C_{ox}}}}{{2L}}{\left( {{V_{GS}} - {V_{th}}} \right)^2}\)

In the linear region of operation, the current is given by:

\({I_D} = {\mu _n}{C_{ox}} \times \frac{W}{L}\left[ {\left( {{V_{GS}} - {V_T}} \right){V_{DS}} - \frac{{V_{DS}^2}}{2}} \right]\)

W = Width of the Gate

Cox = Oxide Capacitance

μ = Mobility of the carrier

L = Channel Length

Vth = Threshold voltage

The transconductance of a MOSFET is defined as the change in drain current(ID) with respect to the corresponding change in gate voltage (VGS), i.e. 

\({g_m} = \frac{{\partial {I_D}}}{{\partial {V_{GS}}}}\)

\(g_m = \frac{{W{μ _x}{C_{ox}}}}{{L}}{\left( {{V_{GS}} - {V_{th}}} \right)}\)

Trans-conductance gm for the linear region will be:

\({g_{m\left( {linear} \right)}} = \frac{{\partial {I_D}}}{{\partial {V_{GS}}}} = \frac{{{\mu _n}{C_{ox}}W}}{L} \times {V_{DS}}\)