Studying fluctuations theory in Hopfield model I found this formula for the derivative of an observable:
$\frac{d\left<O \right>}{ds} = \beta\sqrt{\alpha} \sum_{ab} \left < O \xi_{a, b} \eta_{a, b} \right>_s - s \sum_{a} \left< O \xi_{a, s+1 } \eta_{a, s+1} \right>_s + \frac{s(s+1)}{2} \left < O \xi_{s+1,s+2} \eta_{s+1,s+2}\right>_s \\$
I can't find any reference for the proof of this formula. Everywhere there's written that are just long and boring calculations. Can anyone give me a reference where I can find the full derivation of the formula. Thanks!