There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ log_{lg(x)}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( log_{lg(x)}^{2}\right)}{dx}\\=&(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})\\=&\frac{-log_{lg(x)}^{2}}{xln{10}ln(lg(x))lg(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-log_{lg(x)}^{2}}{xln{10}ln(lg(x))lg(x)}\right)}{dx}\\=&\frac{--log_{lg(x)}^{2}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{xln{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{2}*-0}{xln^{2}{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{2}*-1}{xln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{log_{lg(x)}^{2}*-1}{xln{10}ln(lg(x))lg^{2}(x)ln{10}(x)}\\=&\frac{log_{lg(x)}^{2}}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{2log_{lg(x)}^{2}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{2}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{log_{lg(x)}^{2}}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{2log_{lg(x)}^{2}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{2}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{-2log_{lg(x)}^{2}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{2}*-0}{x^{2}ln^{2}{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{2}*-1}{x^{2}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{log_{lg(x)}^{2}*-1}{x^{2}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{lg(x)}^{2}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{2}*-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{2}*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{2log_{lg(x)}^{2}*-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{2}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{2}*-1}{x^{2}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{2}*-2*0}{x^{2}ln(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{2}*-2}{x^{2}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)}\\=&\frac{-2log_{lg(x)}^{2}}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{6log_{lg(x)}^{2}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{2}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{2}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{2}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{2}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{2}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2log_{lg(x)}^{2}}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{6log_{lg(x)}^{2}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{2}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{2}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{2}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{2}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{2}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\right)}{dx}\\=&\frac{-2*-3log_{lg(x)}^{2}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{2(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{2}*-0}{x^{3}ln^{2}{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{2}*-1}{x^{3}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{2log_{lg(x)}^{2}*-1}{x^{3}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{2}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{2}*-2*0}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{2}*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} - \frac{6log_{lg(x)}^{2}*-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} - \frac{3*-3log_{lg(x)}^{2}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{2}*-1}{x^{3}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{2}*-2*0}{x^{3}ln(lg(x))ln^{3}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{2}*-2}{x^{3}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{2}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{2}*-3*0}{x^{3}ln^{4}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{2}*-3}{x^{3}ln^{3}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{6log_{lg(x)}^{2}*-3}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{4*-3log_{lg(x)}^{2}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{2}*-2}{x^{3}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{2}*-3*0}{x^{3}ln^{2}(lg(x))ln^{4}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{2}*-3}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{2}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{2}*-3*0}{x^{3}ln^{4}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{2}*-2}{x^{3}ln^{3}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{2}*-3}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{2}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(0)}{(2)} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{2}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{2}*-3*0}{x^{3}ln^{4}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{2}*-1}{x^{3}ln^{3}{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{2}*-3}{x^{3}ln^{3}{10}ln(lg(x))lg^{4}(x)ln{10}(x)}\\=&\frac{6log_{lg(x)}^{2}}{x^{4}ln{10}ln(lg(x))lg(x)} + \frac{22log_{lg(x)}^{2}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{11log_{lg(x)}^{2}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{36log_{lg(x)}^{2}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{24log_{lg(x)}^{2}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{12log_{lg(x)}^{2}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{12log_{lg(x)}^{2}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} + \frac{24log_{lg(x)}^{2}}{x^{4}ln^{4}{10}ln^{4}(lg(x))lg^{4}(x)} + \frac{18log_{lg(x)}^{2}}{x^{4}ln^{3}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{18log_{lg(x)}^{2}}{x^{4}ln^{4}{10}ln^{3}(lg(x))lg^{4}(x)} + \frac{16log_{lg(x)}^{2}}{x^{4}ln^{4}{10}ln^{2}(lg(x))lg^{4}(x)} + \frac{6log_{lg(x)}^{2}}{x^{4}ln^{2}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{6log_{lg(x)}^{2}}{x^{4}ln(lg(x))ln^{4}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]

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