There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ ln(\frac{({e}^{x})}{({e}^{x} - 1)})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = ln(\frac{{e}^{x}}{({e}^{x} - 1)})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(\frac{{e}^{x}}{({e}^{x} - 1)})\right)}{dx}\\=&\frac{((\frac{-(({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)})) + 0)}{({e}^{x} - 1)^{2}}){e}^{x} + \frac{({e}^{x}((1)ln(e) + \frac{(x)(0)}{(e)}))}{({e}^{x} - 1)})}{(\frac{{e}^{x}}{({e}^{x} - 1)})}\\=&\frac{-{e}^{x}}{({e}^{x} - 1)} + 1\\ \end{split}\end{equation} \]

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