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

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