There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{log_{x}^{ln(x)}}{ln(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{log_{x}^{ln(x)}}{ln(x)}\right)}{dx}\\=&\frac{(\frac{(\frac{(\frac{1}{(x)})}{(ln(x))} - \frac{(1)log_{x}^{ln(x)}}{(x)})}{(ln(x))})}{ln(x)} + \frac{log_{x}^{ln(x)}*-1}{ln^{2}(x)(x)}\\=&\frac{1}{xln^{3}(x)} - \frac{2log_{x}^{ln(x)}}{xln^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln^{3}(x)} - \frac{2log_{x}^{ln(x)}}{xln^{2}(x)}\right)}{dx}\\=&\frac{-1}{x^{2}ln^{3}(x)} + \frac{-3}{xln^{4}(x)(x)} - \frac{2*-log_{x}^{ln(x)}}{x^{2}ln^{2}(x)} - \frac{2(\frac{(\frac{(\frac{1}{(x)})}{(ln(x))} - \frac{(1)log_{x}^{ln(x)}}{(x)})}{(ln(x))})}{xln^{2}(x)} - \frac{2log_{x}^{ln(x)}*-2}{xln^{3}(x)(x)}\\=&\frac{-1}{x^{2}ln^{3}(x)} - \frac{5}{x^{2}ln^{4}(x)} + \frac{2log_{x}^{ln(x)}}{x^{2}ln^{2}(x)} + \frac{6log_{x}^{ln(x)}}{x^{2}ln^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln^{3}(x)} - \frac{5}{x^{2}ln^{4}(x)} + \frac{2log_{x}^{ln(x)}}{x^{2}ln^{2}(x)} + \frac{6log_{x}^{ln(x)}}{x^{2}ln^{3}(x)}\right)}{dx}\\=&\frac{--2}{x^{3}ln^{3}(x)} - \frac{-3}{x^{2}ln^{4}(x)(x)} - \frac{5*-2}{x^{3}ln^{4}(x)} - \frac{5*-4}{x^{2}ln^{5}(x)(x)} + \frac{2*-2log_{x}^{ln(x)}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(\frac{1}{(x)})}{(ln(x))} - \frac{(1)log_{x}^{ln(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{ln(x)}*-2}{x^{2}ln^{3}(x)(x)} + \frac{6*-2log_{x}^{ln(x)}}{x^{3}ln^{3}(x)} + \frac{6(\frac{(\frac{(\frac{1}{(x)})}{(ln(x))} - \frac{(1)log_{x}^{ln(x)}}{(x)})}{(ln(x))})}{x^{2}ln^{3}(x)} + \frac{6log_{x}^{ln(x)}*-3}{x^{2}ln^{4}(x)(x)}\\=&\frac{2}{x^{3}ln^{3}(x)} + \frac{15}{x^{3}ln^{4}(x)} + \frac{26}{x^{3}ln^{5}(x)} - \frac{4log_{x}^{ln(x)}}{x^{3}ln^{2}(x)} - \frac{18log_{x}^{ln(x)}}{x^{3}ln^{3}(x)} - \frac{24log_{x}^{ln(x)}}{x^{3}ln^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln^{3}(x)} + \frac{15}{x^{3}ln^{4}(x)} + \frac{26}{x^{3}ln^{5}(x)} - \frac{4log_{x}^{ln(x)}}{x^{3}ln^{2}(x)} - \frac{18log_{x}^{ln(x)}}{x^{3}ln^{3}(x)} - \frac{24log_{x}^{ln(x)}}{x^{3}ln^{4}(x)}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln^{3}(x)} + \frac{2*-3}{x^{3}ln^{4}(x)(x)} + \frac{15*-3}{x^{4}ln^{4}(x)} + \frac{15*-4}{x^{3}ln^{5}(x)(x)} + \frac{26*-3}{x^{4}ln^{5}(x)} + \frac{26*-5}{x^{3}ln^{6}(x)(x)} - \frac{4*-3log_{x}^{ln(x)}}{x^{4}ln^{2}(x)} - \frac{4(\frac{(\frac{(\frac{1}{(x)})}{(ln(x))} - \frac{(1)log_{x}^{ln(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{4log_{x}^{ln(x)}*-2}{x^{3}ln^{3}(x)(x)} - \frac{18*-3log_{x}^{ln(x)}}{x^{4}ln^{3}(x)} - \frac{18(\frac{(\frac{(\frac{1}{(x)})}{(ln(x))} - \frac{(1)log_{x}^{ln(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{18log_{x}^{ln(x)}*-3}{x^{3}ln^{4}(x)(x)} - \frac{24*-3log_{x}^{ln(x)}}{x^{4}ln^{4}(x)} - \frac{24(\frac{(\frac{(\frac{1}{(x)})}{(ln(x))} - \frac{(1)log_{x}^{ln(x)}}{(x)})}{(ln(x))})}{x^{3}ln^{4}(x)} - \frac{24log_{x}^{ln(x)}*-4}{x^{3}ln^{5}(x)(x)}\\=&\frac{-6}{x^{4}ln^{3}(x)} - \frac{55}{x^{4}ln^{4}(x)} - \frac{156}{x^{4}ln^{5}(x)} - \frac{154}{x^{4}ln^{6}(x)} + \frac{12log_{x}^{ln(x)}}{x^{4}ln^{2}(x)} + \frac{66log_{x}^{ln(x)}}{x^{4}ln^{3}(x)} + \frac{144log_{x}^{ln(x)}}{x^{4}ln^{4}(x)} + \frac{120log_{x}^{ln(x)}}{x^{4}ln^{5}(x)}\\ \end{split}\end{equation} \]

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