There are 1 questions in this calculation: for each question, the 2 derivative of T is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ ln(({(2cosh(\frac{j}{(kT)}))}^{N}))\ with\ respect\ to\ T：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln((2cosh(\frac{j}{kT}))^{N})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln((2cosh(\frac{j}{kT}))^{N})\right)}{dT}\\=&\frac{((2cosh(\frac{j}{kT}))^{N}((0)ln(2cosh(\frac{j}{kT})) + \frac{(N)(\frac{2sinh(\frac{j}{kT})j*-1}{kT^{2}})}{(2cosh(\frac{j}{kT}))}))}{((2cosh(\frac{j}{kT}))^{N})}\\=&\frac{-jNsinh(\frac{j}{kT})}{kT^{2}cosh(\frac{j}{kT})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-jNsinh(\frac{j}{kT})}{kT^{2}cosh(\frac{j}{kT})}\right)}{dT}\\=&\frac{-jN*-2sinh(\frac{j}{kT})}{kT^{3}cosh(\frac{j}{kT})} - \frac{jNcosh(\frac{j}{kT})j*-1}{kT^{2}kT^{2}cosh(\frac{j}{kT})} - \frac{jNsinh(\frac{j}{kT})*-sinh(\frac{j}{kT})j*-1}{kT^{2}cosh^{2}(\frac{j}{kT})kT^{2}}\\=&\frac{2jNsinh(\frac{j}{kT})}{kT^{3}cosh(\frac{j}{kT})} - \frac{j^{2}Nsinh^{2}(\frac{j}{kT})}{k^{2}T^{4}cosh^{2}(\frac{j}{kT})} + \frac{j^{2}N}{k^{2}T^{4}}\\ \end{split}\end{equation} \]

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