This is a bit of a long question so bear with me. In chapter 3 of Razavi's Design of Analog CMOS Integrated Circuits, he introduces the CS stage with source degeneration. He draws the small-signal model and says that the output resistance is as follows (Eqn 3.65):
That makes complete sense to me and I am able to derive that via the small-signal model. Next, Razavi provides another technique to derive the same small-signal output resistance without drawing the small-signal model but rather incorporating it with the large-signal drawing as a means to quickly inspect circuits and gain some intuition. To do this, he applies a ΔV and measures ΔI as shown below,
I understand how he has transitioned from Fig 3.30(a) to (b) and to (c). My confusion is regarding the final step where he calculates the output resistance itself,
I can understand what he is doing but what I don't understand is that as soon as he simplifies the circuit to Fig. 3.30(c), we have a circuit that comprises only of resistors.
Why can you not simply at this stage (Fig 3.30(c)) just write that, $$ R_{out} = r_o + Rs||\frac{1}{g_m + g_{mb}}$$
Why does this not give the same result as Eqn 3.65?