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

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