MOSFET - Derivation of I/V Characteristics - Small Vds
It's just some math:

\$I_D=k[v_{ov}v_{ds}-\frac{1}{2}v_{ds}^2] \$

\$ \ \  \ \ \ =\frac{1}{2}k.v_{ds}[2v_{ov}-v_{ds}] \$

Assuming the MOSFET in deep triode region, and 2v_ov >> vds, the expression for ID becomes,

\$I_D=\frac{1}{2}k.v_{ds}2v_{ov}\$

\$ \ \ \ \ =k.v_{ds}v_{ov}\$

Here \$v_{ov}=v_{gs} - v_{th}\$ and \$k=\mu _nc_{ox}w/l\$. The assumption 2v_ov >> vds has to be satisfied in order for the transistor to act as a linear resistor in the deep tride region, with the resistance being \$\frac{1}{kv_{ov}}\$.