Derivative of the partition function Help

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To demonstrate that the average energy is given by the formula -(1/Z)(dZ/dBeta) = -(d/dBeta)Ln(Z), one must correctly apply the chain rule in calculus. The partition function Z is defined as the sum over states, Z = sum over s of { e^(beta*E(s)) }, where Beta = 1/kT. The derivative of ln(Z) is indeed 1/Z(dZ/dBeta), which connects the two expressions. Careful attention to the differentiation process is necessary to avoid errors. Understanding these relationships is crucial for mastering statistical mechanics.
vuser88
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i need to show that the average value of the energy is -(1/Z)(dZ/dBeta)= -(d/dBeta)Ln(Z)

where Z is the partition function i know how to do the first part, i don't know how to show this is equal to the derivative w/ respect to beta of lnZ. i think my math is wrong when taking Ln(Z)

Beta = 1/kT
Z= sum over s of { e^ (beta*E(s)) }
any suggestions,

ps i do have the solution from cramster but i don't want to simply copy it because then i will never learn anything
 
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im pretty sure sure this is the chain rule, but it dosent work out when i actually do it step by step
 
Are you aware that the derivative of ln(Z) is 1/Z?
 
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