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

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