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

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