There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{ln(x)}{(e^{x}(x - 1))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{ln(x)}{(xe^{x} - e^{x})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ln(x)}{(xe^{x} - e^{x})}\right)}{dx}\\=&(\frac{-(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{2}})ln(x) + \frac{1}{(xe^{x} - e^{x})(x)}\\=&\frac{-xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} + \frac{1}{(xe^{x} - e^{x})x}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} + \frac{1}{(xe^{x} - e^{x})x}\right)}{dx}\\=&-(\frac{-2(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{3}})xe^{x}ln(x) - \frac{e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}}{(xe^{x} - e^{x})^{2}(x)} + \frac{(\frac{-(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{2}})}{x} + \frac{-1}{(xe^{x} - e^{x})x^{2}}\\=&\frac{2x^{2}e^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} - \frac{e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{2e^{x}}{(xe^{x} - e^{x})^{2}} - \frac{1}{(xe^{x} - e^{x})x^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2x^{2}e^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} - \frac{e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{2e^{x}}{(xe^{x} - e^{x})^{2}} - \frac{1}{(xe^{x} - e^{x})x^{2}}\right)}{dx}\\=&2(\frac{-3(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{4}})x^{2}e^{{x}*{2}}ln(x) + \frac{2*2xe^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{2x^{2}*2e^{x}e^{x}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{2x^{2}e^{{x}*{2}}}{(xe^{x} - e^{x})^{3}(x)} - (\frac{-2(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{3}})e^{x}ln(x) - \frac{e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{e^{x}}{(xe^{x} - e^{x})^{2}(x)} - (\frac{-2(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{3}})xe^{x}ln(x) - \frac{e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}}{(xe^{x} - e^{x})^{2}(x)} - 2(\frac{-2(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{3}})e^{x} - \frac{2e^{x}}{(xe^{x} - e^{x})^{2}} - \frac{(\frac{-(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{2}})}{x^{2}} - \frac{-2}{(xe^{x} - e^{x})x^{3}}\\=&\frac{-6x^{3}e^{{x}*{3}}ln(x)}{(xe^{x} - e^{x})^{4}} + \frac{6xe^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{6x^{2}e^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} - \frac{xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{2e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} + \frac{6xe^{{x}*{2}}}{(xe^{x} - e^{x})^{3}} - \frac{3e^{x}}{(xe^{x} - e^{x})^{2}} + \frac{2}{(xe^{x} - e^{x})x^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6x^{3}e^{{x}*{3}}ln(x)}{(xe^{x} - e^{x})^{4}} + \frac{6xe^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{6x^{2}e^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} - \frac{xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{2e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} + \frac{6xe^{{x}*{2}}}{(xe^{x} - e^{x})^{3}} - \frac{3e^{x}}{(xe^{x} - e^{x})^{2}} + \frac{2}{(xe^{x} - e^{x})x^{3}}\right)}{dx}\\=&-6(\frac{-4(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{5}})x^{3}e^{{x}*{3}}ln(x) - \frac{6*3x^{2}e^{{x}*{3}}ln(x)}{(xe^{x} - e^{x})^{4}} - \frac{6x^{3}*3e^{{x}*{2}}e^{x}ln(x)}{(xe^{x} - e^{x})^{4}} - \frac{6x^{3}e^{{x}*{3}}}{(xe^{x} - e^{x})^{4}(x)} + 6(\frac{-3(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{4}})xe^{{x}*{2}}ln(x) + \frac{6e^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{6x*2e^{x}e^{x}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{6xe^{{x}*{2}}}{(xe^{x} - e^{x})^{3}(x)} + 6(\frac{-3(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{4}})x^{2}e^{{x}*{2}}ln(x) + \frac{6*2xe^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{6x^{2}*2e^{x}e^{x}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{6x^{2}e^{{x}*{2}}}{(xe^{x} - e^{x})^{3}(x)} - (\frac{-2(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{3}})xe^{x}ln(x) - \frac{e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}}{(xe^{x} - e^{x})^{2}(x)} - 2(\frac{-2(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{3}})e^{x}ln(x) - \frac{2e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{2e^{x}}{(xe^{x} - e^{x})^{2}(x)} + 6(\frac{-3(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{4}})xe^{{x}*{2}} + \frac{6e^{{x}*{2}}}{(xe^{x} - e^{x})^{3}} + \frac{6x*2e^{x}e^{x}}{(xe^{x} - e^{x})^{3}} - 3(\frac{-2(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{3}})e^{x} - \frac{3e^{x}}{(xe^{x} - e^{x})^{2}} + \frac{2(\frac{-(e^{x} + xe^{x} - e^{x})}{(xe^{x} - e^{x})^{2}})}{x^{3}} + \frac{2*-3}{(xe^{x} - e^{x})x^{4}}\\=&\frac{24x^{4}e^{{x}*{4}}ln(x)}{(xe^{x} - e^{x})^{5}} - \frac{36x^{2}e^{{x}*{3}}ln(x)}{(xe^{x} - e^{x})^{4}} - \frac{36x^{3}e^{{x}*{3}}ln(x)}{(xe^{x} - e^{x})^{4}} + \frac{28xe^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{6e^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} + \frac{14x^{2}e^{{x}*{2}}ln(x)}{(xe^{x} - e^{x})^{3}} - \frac{3e^{x}ln(x)}{(xe^{x} - e^{x})^{2}} - \frac{xe^{x}ln(x)}{(xe^{x} - e^{x})^{2}} + \frac{24xe^{{x}*{2}}}{(xe^{x} - e^{x})^{3}} + \frac{12e^{{x}*{2}}}{(xe^{x} - e^{x})^{3}} - \frac{24x^{2}e^{{x}*{3}}}{(xe^{x} - e^{x})^{4}} - \frac{4e^{x}}{(xe^{x} - e^{x})^{2}} - \frac{2e^{x}}{(xe^{x} - e^{x})^{2}x} - \frac{2e^{x}}{(xe^{x} - e^{x})^{2}x^{2}} - \frac{6}{(xe^{x} - e^{x})x^{4}}\\ \end{split}\end{equation} \]

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