There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{x}^{x}^{\frac{1}{5}}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{x}^{x}^{\frac{1}{5}}}\right)}{dx}\\=&(\frac{(\frac{(1)}{(x}^{\frac{1}{5}})} - \frac{(1)log_{x}^{x}^{\frac{1}{5}}}}{(x)})}{(ln(x))})\\=&\frac{-log_{x}^{x}^{\frac{1}{5}}}}{xln(x)} + \frac{1}{x, \frac{1}{5})ln(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-log_{x}^{x}^{\frac{1}{5}}}}{xln(x)} + \frac{1}{x, \frac{1}{5})ln(x)}\right)}{dx}\\=&\frac{--log_{x}^{x}^{\frac{1}{5}}}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(1)}{(x}^{\frac{1}{5}})} - \frac{(1)log_{x}^{x}^{\frac{1}{5}}}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{x}^{\frac{1}{5}}}*-1}{xln^{2}(x)(x)} + \frac{-1}{x^{2}ln(x)} + \frac{-1}{x, \frac{1}{5})ln^{2}(x)(x)}\\=&\frac{log_{x}^{x}^{\frac{1}{5}}}}{x^{2}ln(x)} + \frac{2log_{x}^{x}^{\frac{1}{5}}}}{x^{2}ln^{2}(x)} - \frac{1}{x, \frac{1}{5})xln^{2}(x)} - \frac{1}{x^{2}ln(x)} - \frac{1}{xx, \frac{1}{5})ln^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{log_{x}^{x}^{\frac{1}{5}}}}{x^{2}ln(x)} + \frac{2log_{x}^{x}^{\frac{1}{5}}}}{x^{2}ln^{2}(x)} - \frac{1}{x, \frac{1}{5})xln^{2}(x)} - \frac{1}{x^{2}ln(x)} - \frac{1}{xx, \frac{1}{5})ln^{2}(x)}\right)}{dx}\\=&\frac{-2log_{x}^{x}^{\frac{1}{5}}}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(1)}{(x}^{\frac{1}{5}})} - \frac{(1)log_{x}^{x}^{\frac{1}{5}}}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{x}^{\frac{1}{5}}}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{x}^{\frac{1}{5}}}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(1)}{(x}^{\frac{1}{5}})} - \frac{(1)log_{x}^{x}^{\frac{1}{5}}}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{x}^{\frac{1}{5}}}*-2}{x^{2}ln^{3}(x)(x)} - \frac{-1}{x^{2}xln^{2}(x)} - \frac{-1}{x, \frac{1}{5})x^{2}ln^{2}(x)} - \frac{-2}{x, \frac{1}{5})xln^{3}(x)(x)} - \frac{-2}{x^{3}ln(x)} - \frac{-1}{x^{2}ln^{2}(x)(x)} - \frac{-1}{x^{2}x, \frac{1}{5})ln^{2}(x)} - \frac{-1}{xx^{2}ln^{2}(x)} - \frac{-2}{xx, \frac{1}{5})ln^{3}(x)(x)}\\=&\frac{-2log_{x}^{x}^{\frac{1}{5}}}}{x^{3}ln(x)} - \frac{6log_{x}^{x}^{\frac{1}{5}}}}{x^{3}ln^{2}(x)} + \frac{2}{x, \frac{1}{5})x^{2}ln^{2}(x)} - \frac{6log_{x}^{x}^{\frac{1}{5}}}}{x^{3}ln^{3}(x)} + \frac{4}{x, \frac{1}{5})x^{2}ln^{3}(x)} + \frac{3}{x^{3}ln^{2}(x)} + \frac{2}{x^{2}x, \frac{1}{5})ln^{3}(x)} + \frac{2}{x^{3}ln(x)} + \frac{1}{x^{2}x, \frac{1}{5})ln^{2}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2log_{x}^{x}^{\frac{1}{5}}}}{x^{3}ln(x)} - \frac{6log_{x}^{x}^{\frac{1}{5}}}}{x^{3}ln^{2}(x)} + \frac{2}{x, \frac{1}{5})x^{2}ln^{2}(x)} - \frac{6log_{x}^{x}^{\frac{1}{5}}}}{x^{3}ln^{3}(x)} + \frac{4}{x, \frac{1}{5})x^{2}ln^{3}(x)} + \frac{3}{x^{3}ln^{2}(x)} + \frac{2}{x^{2}x, \frac{1}{5})ln^{3}(x)} + \frac{2}{x^{3}ln(x)} + \frac{1}{x^{2}x, \frac{1}{5})ln^{2}(x)}\right)}{dx}\\=&\frac{-2*-3log_{x}^{x}^{\frac{1}{5}}}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(1)}{(x}^{\frac{1}{5}})} - \frac{(1)log_{x}^{x}^{\frac{1}{5}}}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{x}^{\frac{1}{5}}}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{x}^{\frac{1}{5}}}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(1)}{(x}^{\frac{1}{5}})} - \frac{(1)log_{x}^{x}^{\frac{1}{5}}}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{x}^{\frac{1}{5}}}*-2}{x^{3}ln^{3}(x)(x)} + \frac{2*-1}{x^{2}x^{2}ln^{2}(x)} + \frac{2*-2}{x, \frac{1}{5})x^{3}ln^{2}(x)} + \frac{2*-2}{x, \frac{1}{5})x^{2}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{x}^{\frac{1}{5}}}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(1)}{(x}^{\frac{1}{5}})} - \frac{(1)log_{x}^{x}^{\frac{1}{5}}}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{x}^{\frac{1}{5}}}*-3}{x^{3}ln^{4}(x)(x)} + \frac{4*-1}{x^{2}x^{2}ln^{3}(x)} + \frac{4*-2}{x, \frac{1}{5})x^{3}ln^{3}(x)} + \frac{4*-3}{x, \frac{1}{5})x^{2}ln^{4}(x)(x)} + \frac{3*-3}{x^{4}ln^{2}(x)} + \frac{3*-2}{x^{3}ln^{3}(x)(x)} + \frac{2*-2}{x^{3}x, \frac{1}{5})ln^{3}(x)} + \frac{2*-1}{x^{2}x^{2}ln^{3}(x)} + \frac{2*-3}{x^{2}x, \frac{1}{5})ln^{4}(x)(x)} + \frac{2*-3}{x^{4}ln(x)} + \frac{2*-1}{x^{3}ln^{2}(x)(x)} + \frac{-2}{x^{3}x, \frac{1}{5})ln^{2}(x)} + \frac{-1}{x^{2}x^{2}ln^{2}(x)} + \frac{-2}{x^{2}x, \frac{1}{5})ln^{3}(x)(x)}\\=&\frac{6log_{x}^{x}^{\frac{1}{5}}}}{x^{4}ln(x)} + \frac{22log_{x}^{x}^{\frac{1}{5}}}}{x^{4}ln^{2}(x)} - \frac{6}{x, \frac{1}{5})x^{3}ln^{2}(x)} + \frac{36log_{x}^{x}^{\frac{1}{5}}}}{x^{4}ln^{3}(x)} - \frac{16}{x, \frac{1}{5})x^{3}ln^{3}(x)} - \frac{14}{x^{4}ln^{2}(x)} - \frac{8}{x^{3}x, \frac{1}{5})ln^{3}(x)} + \frac{24log_{x}^{x}^{\frac{1}{5}}}}{x^{4}ln^{4}(x)} - \frac{12}{x, \frac{1}{5})x^{3}ln^{4}(x)} - \frac{12}{x^{4}ln^{3}(x)} - \frac{12}{x^{3}x, \frac{1}{5})ln^{4}(x)} - \frac{6}{x^{4}ln(x)} - \frac{2}{x^{3}x, \frac{1}{5})ln^{2}(x)}\\ \end{split}\end{equation} \]

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