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

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