Derivation of Average Square Energy Fluctuation in a Canonical System

[phun_physics]

TL;DR Summary

I am currently reading through Chandler's Introduction to Modern Statistical Mechanics and would like insight into this derivation

The canonical ( Boltzmann) distribution law for a canonical system is described the probability of state $v$ by $P_v = Q^{-1} e^{-\beta E_v}$ where $Q^{-1}$ is the normalization constant of $\sum_v P_v = 1$ and therefore $Q = \sum_{v}e^{-\beta E_v}$. Chandler then derives $\langle( \delta E_v)^2 \rangle$ in the attachment.

I am confused on how he went from these steps: $Q^{-1}(\partial^2 Q / \partial \beta^2)_{N, V} - Q^{-2}(\partial Q / \partial \beta)^{2}_{N,V}$ = $(\partial ^2 lnQ / \partial \beta^2)_{N,V}$

Any help would be extremely appreciated! Thanks!

 

[DrClaude]

I am confused on how he went from these steps: $Q^{-1}(\partial^2 Q / \partial \beta^2)_{N, V} - Q^{-2}(\partial Q / \partial \beta)^{2}_{N,V}$ = $(\partial ^2 lnQ / \partial \beta^2)_{N,V}$

Do it backwards. Start from $(\partial ^2 \ln Q / \partial \beta^2)_{N,V}$ and calculate the derivatives.