I had a quick question on deriving voltage gain of a common-source amp with source degeneration. Please see attached PDF file. I derived the voltage gain using the regular method (method 1) and by inspection (method2). I get different answers from the 2 methods, although upon plugging in the numbers the answers are numerically about the same (same up to the second decimal). I am unable to algebraically reduce the symbolic results to show they are equal.

Method 1 in your PDF is exact but it doesn't lend itself to intuitive understanding. Method 2 is easy to understand intuitively but it isn't exact. Yes, as you noted the results are about the same.

Why the difference? When you look at the resistance looking into the source of the MOSFET it isn't exactly 1/gm as specified in Method 2 but rather

(1 + Ro/ro)/(gm +2/ro + Ro/ro^2)

If ro >> Ro then the resistance looking into the source is 1/gm and the two methods give the same results.

In later printings of the book I changed the more advanced discussion of the resistance looking into the source with a very large load Ro, see pages 688-690 here, to keep things more general. Specifically see Eq. (21.81) and Fig. 21.37.

For the question at the bottom of the PDF, superposition allows you to look at sources separately so it is fine to short-out the input source when determining the resistance looking into the drain. For exact analysis make sure you use the same circuit for all analyses. Don't lump the resistance looking into the drain into a single resistor connected to the output, like we do for the fast/intuitive analysis, and then try to calculate the exact gain.

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