There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{e^{x}}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{e^{x}}^{x}\right)}{dx}\\=&(\frac{(\frac{(1)}{(x)} - \frac{(e^{x})log_{e^{x}}^{x}}{(e^{x})})}{(ln(e^{x}))})\\=&\frac{1}{xln(e^{x})} - \frac{log_{e^{x}}^{x}}{ln(e^{x})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln(e^{x})} - \frac{log_{e^{x}}^{x}}{ln(e^{x})}\right)}{dx}\\=&\frac{-1}{x^{2}ln(e^{x})} + \frac{-e^{x}}{xln^{2}(e^{x})(e^{x})} - \frac{(\frac{(\frac{(1)}{(x)} - \frac{(e^{x})log_{e^{x}}^{x}}{(e^{x})})}{(ln(e^{x}))})}{ln(e^{x})} - \frac{log_{e^{x}}^{x}*-e^{x}}{ln^{2}(e^{x})(e^{x})}\\=&\frac{-1}{x^{2}ln(e^{x})} - \frac{2}{xln^{2}(e^{x})} + \frac{2log_{e^{x}}^{x}}{ln^{2}(e^{x})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln(e^{x})} - \frac{2}{xln^{2}(e^{x})} + \frac{2log_{e^{x}}^{x}}{ln^{2}(e^{x})}\right)}{dx}\\=&\frac{--2}{x^{3}ln(e^{x})} - \frac{-e^{x}}{x^{2}ln^{2}(e^{x})(e^{x})} - \frac{2*-1}{x^{2}ln^{2}(e^{x})} - \frac{2*-2e^{x}}{xln^{3}(e^{x})(e^{x})} + \frac{2(\frac{(\frac{(1)}{(x)} - \frac{(e^{x})log_{e^{x}}^{x}}{(e^{x})})}{(ln(e^{x}))})}{ln^{2}(e^{x})} + \frac{2log_{e^{x}}^{x}*-2e^{x}}{ln^{3}(e^{x})(e^{x})}\\=&\frac{2}{x^{3}ln(e^{x})} + \frac{3}{x^{2}ln^{2}(e^{x})} + \frac{6}{xln^{3}(e^{x})} - \frac{6log_{e^{x}}^{x}}{ln^{3}(e^{x})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln(e^{x})} + \frac{3}{x^{2}ln^{2}(e^{x})} + \frac{6}{xln^{3}(e^{x})} - \frac{6log_{e^{x}}^{x}}{ln^{3}(e^{x})}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln(e^{x})} + \frac{2*-e^{x}}{x^{3}ln^{2}(e^{x})(e^{x})} + \frac{3*-2}{x^{3}ln^{2}(e^{x})} + \frac{3*-2e^{x}}{x^{2}ln^{3}(e^{x})(e^{x})} + \frac{6*-1}{x^{2}ln^{3}(e^{x})} + \frac{6*-3e^{x}}{xln^{4}(e^{x})(e^{x})} - \frac{6(\frac{(\frac{(1)}{(x)} - \frac{(e^{x})log_{e^{x}}^{x}}{(e^{x})})}{(ln(e^{x}))})}{ln^{3}(e^{x})} - \frac{6log_{e^{x}}^{x}*-3e^{x}}{ln^{4}(e^{x})(e^{x})}\\=&\frac{-6}{x^{4}ln(e^{x})} - \frac{8}{x^{3}ln^{2}(e^{x})} - \frac{12}{x^{2}ln^{3}(e^{x})} - \frac{24}{xln^{4}(e^{x})} + \frac{24log_{e^{x}}^{x}}{ln^{4}(e^{x})}\\ \end{split}\end{equation} \]

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