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1. Z = summation ( exp ( - B*E(s)) ) where the sum is over s

d/dB of ln(Z) = d/dB (ln (exp(-BEo) + exp(-BE1) + ... exp(-BEn))

= (exp(-BEo) + exp(-BE1) + ... exp(-BEn))^-1 +

(-E0*exp(-BEo) + -E1*exp(-BE1) + ... -En*exp(-BEn))

= summation ( E(s) * exp(-B*E(s)) / summation ( exp(-B*E(s))

which is also the average value of E when Prob(E(si)) = exp(-BE(si))

2. does d/dT of exp( -E/kT) = -E/k * exp(-E/kT) * -(1/T^2) =

E/k* 1/T^2 * exp(-E/kT) ?

If you're curious, these come up in Boltzmann statistics in thermal physics.