Differential amplifier gain calculation

[dbsamithey]

I derived the gain of a differential amplifier in the attached write-up.  What worked in the end was essentially writing a few voltage loop equations, differentiating with respect to the voltage across one of the tubes, and applying chain rule repeatedly.  Is there a simpler way?

[PRR]

https://www.worldradiohistory.com/Archive-Company-Publications/Philips-Technical-Review/60s/Philips-Technical-Review-1960.pdf    page 345 (PDF page 349)

https://www.worldradiohistory.com/Archive-Company-Publications/Philips-Technical-Review/60s/Philips-Technical-Review-1961.pdf    page 142 (PDF page 147)

Typical Philips thoroughness.

[dbsamithey]

Oh wow these are gold.  Judging by the first full paragraph below fig. 8 in that document from '61, it seems like maybe it's a bit improper (or plain wrong) to not have the screen and cathode of the pentode connected with a capacitor?  I figured that pinning the screen at ac ground would provide the needed stabilization under oscillations of the anode voltage, but maybe that's not right.  It looks like the example circuit in the 6au6 datasheet has bypass caps on both the cathode and screen to ground.  Something to try I guess.  Fortunately, amateur mistakes are forgiven when you're working with 1 MOhm of resistance.