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

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