Mosfet equation: how to get it?
$$\frac{dI_D}{dV_t}=\frac{d}{dV_t}k(V_G-R_SI_D -V_t)^2$$
$$\implies \frac{dI_D}{dV_t} = 2k(V_G-R_SI_D -V_t)(0-R_S\frac{dI_D}{dV_t}-1)$$
$$\implies \frac{dI_D}{dV_t} = 2k(V_G-R_SI_D -V_t)(0-R_S\frac{dI_D}{dV_t}-1) ---(1)$$
Given that:
$$I_D=k(V_G-R_SI_D -V_t)^2$$
$$\implies (V_G-R_SI_D -V_t) = \sqrt{I_D/k} $$
Therefore (1) \$ \implies\$
$$\frac{dI_D}{dV_t} = 2\sqrt{kI_D}(-R_S\frac{dI_D}{dV_t} -1)$$
$$\implies \frac{dI_D}{dV_t} +2\sqrt{kI_D}R_S\frac{dI_D}{dV_t} = -2\sqrt{kI_D} $$
$$\implies\frac{dI_D}{dV_t}(1+2\sqrt{kI_D}R_S) = -2\sqrt{kI_D}  $$
$$\implies \frac{dI_D}{dV_t} =\frac{-2\sqrt{kI_D}}{1+2R_S\sqrt{kI_D}} $$