There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ x(e^{x} - 1) - \frac{1}{(2{x}^{2})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( xe^{x} - x - \frac{\frac{1}{2}}{x^{2}}\right)}{dx}\\=&e^{x} + xe^{x} - 1 - \frac{\frac{1}{2}*-2}{x^{3}}\\=&e^{x} + xe^{x} + \frac{1}{x^{3}} - 1\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( e^{x} + xe^{x} + \frac{1}{x^{3}} - 1\right)}{dx}\\=&e^{x} + e^{x} + xe^{x} + \frac{-3}{x^{4}} + 0\\=&2e^{x} + xe^{x} - \frac{3}{x^{4}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( 2e^{x} + xe^{x} - \frac{3}{x^{4}}\right)}{dx}\\=&2e^{x} + e^{x} + xe^{x} - \frac{3*-4}{x^{5}}\\=&3e^{x} + xe^{x} + \frac{12}{x^{5}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( 3e^{x} + xe^{x} + \frac{12}{x^{5}}\right)}{dx}\\=&3e^{x} + e^{x} + xe^{x} + \frac{12*-5}{x^{6}}\\=&4e^{x} + xe^{x} - \frac{60}{x^{6}}\\ \end{split}\end{equation} \]

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