There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{e^{x}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{e^{x}}\right)}{dx}\\=&(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(1)log_{x}^{e^{x}}}{(x)})}{(ln(x))})\\=&\frac{1}{ln(x)} - \frac{log_{x}^{e^{x}}}{xln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{ln(x)} - \frac{log_{x}^{e^{x}}}{xln(x)}\right)}{dx}\\=&\frac{-1}{ln^{2}(x)(x)} - \frac{-log_{x}^{e^{x}}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(1)log_{x}^{e^{x}}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{e^{x}}*-1}{xln^{2}(x)(x)}\\=&\frac{-2}{xln^{2}(x)} + \frac{log_{x}^{e^{x}}}{x^{2}ln(x)} + \frac{2log_{x}^{e^{x}}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2}{xln^{2}(x)} + \frac{log_{x}^{e^{x}}}{x^{2}ln(x)} + \frac{2log_{x}^{e^{x}}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2*-1}{x^{2}ln^{2}(x)} - \frac{2*-2}{xln^{3}(x)(x)} + \frac{-2log_{x}^{e^{x}}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(1)log_{x}^{e^{x}}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{e^{x}}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{e^{x}}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(1)log_{x}^{e^{x}}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{e^{x}}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{3}{x^{2}ln^{2}(x)} + \frac{6}{x^{2}ln^{3}(x)} - \frac{2log_{x}^{e^{x}}}{x^{3}ln(x)} - \frac{6log_{x}^{e^{x}}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{e^{x}}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3}{x^{2}ln^{2}(x)} + \frac{6}{x^{2}ln^{3}(x)} - \frac{2log_{x}^{e^{x}}}{x^{3}ln(x)} - \frac{6log_{x}^{e^{x}}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{e^{x}}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{3*-2}{x^{3}ln^{2}(x)} + \frac{3*-2}{x^{2}ln^{3}(x)(x)} + \frac{6*-2}{x^{3}ln^{3}(x)} + \frac{6*-3}{x^{2}ln^{4}(x)(x)} - \frac{2*-3log_{x}^{e^{x}}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(1)log_{x}^{e^{x}}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{e^{x}}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{e^{x}}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(1)log_{x}^{e^{x}}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{e^{x}}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{e^{x}}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(1)log_{x}^{e^{x}}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{e^{x}}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{-8}{x^{3}ln^{2}(x)} - \frac{24}{x^{3}ln^{3}(x)} - \frac{24}{x^{3}ln^{4}(x)} + \frac{6log_{x}^{e^{x}}}{x^{4}ln(x)} + \frac{22log_{x}^{e^{x}}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{e^{x}}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{e^{x}}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]

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