There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(lg(lg(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(lg(lg(x)))\right)}{dx}\\=&\frac{1}{ln{10}(lg(lg(x)))ln{10}(lg(x))ln{10}(x)}\\=&\frac{1}{xln^{3}{10}lg(x)lg(lg(x))}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xln^{3}{10}lg(x)lg(lg(x))}\right)}{dx}\\=&\frac{-1}{x^{2}ln^{3}{10}lg(x)lg(lg(x))} + \frac{-3*0}{xln^{4}{10}lg(x)lg(lg(x))} + \frac{-1}{xln^{3}{10}lg^{2}(x)ln{10}(x)lg(lg(x))} + \frac{-1}{xln^{3}{10}lg(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)}\\=&\frac{-1}{x^{2}ln^{3}{10}lg(x)lg(lg(x))} - \frac{1}{x^{2}ln^{4}{10}lg^{2}(x)lg(lg(x))} - \frac{1}{x^{2}ln^{5}{10}lg^{2}(x)lg^{2}(lg(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}ln^{3}{10}lg(x)lg(lg(x))} - \frac{1}{x^{2}ln^{4}{10}lg^{2}(x)lg(lg(x))} - \frac{1}{x^{2}ln^{5}{10}lg^{2}(x)lg^{2}(lg(x))}\right)}{dx}\\=&\frac{--2}{x^{3}ln^{3}{10}lg(x)lg(lg(x))} - \frac{-3*0}{x^{2}ln^{4}{10}lg(x)lg(lg(x))} - \frac{-1}{x^{2}ln^{3}{10}lg^{2}(x)ln{10}(x)lg(lg(x))} - \frac{-1}{x^{2}ln^{3}{10}lg(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)} - \frac{-2}{x^{3}ln^{4}{10}lg^{2}(x)lg(lg(x))} - \frac{-4*0}{x^{2}ln^{5}{10}lg^{2}(x)lg(lg(x))} - \frac{-2}{x^{2}ln^{4}{10}lg^{3}(x)ln{10}(x)lg(lg(x))} - \frac{-1}{x^{2}ln^{4}{10}lg^{2}(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)} - \frac{-2}{x^{3}ln^{5}{10}lg^{2}(x)lg^{2}(lg(x))} - \frac{-5*0}{x^{2}ln^{6}{10}lg^{2}(x)lg^{2}(lg(x))} - \frac{-2}{x^{2}ln^{5}{10}lg^{3}(x)ln{10}(x)lg^{2}(lg(x))} - \frac{-2}{x^{2}ln^{5}{10}lg^{2}(x)lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)}\\=&\frac{2}{x^{3}ln^{3}{10}lg(x)lg(lg(x))} + \frac{3}{x^{3}ln^{4}{10}lg^{2}(x)lg(lg(x))} + \frac{3}{x^{3}ln^{5}{10}lg^{2}(x)lg^{2}(lg(x))} + \frac{2}{x^{3}ln^{5}{10}lg^{3}(x)lg(lg(x))} + \frac{3}{x^{3}ln^{6}{10}lg^{3}(x)lg^{2}(lg(x))} + \frac{2}{x^{3}ln^{7}{10}lg^{3}(x)lg^{3}(lg(x))}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}ln^{3}{10}lg(x)lg(lg(x))} + \frac{3}{x^{3}ln^{4}{10}lg^{2}(x)lg(lg(x))} + \frac{3}{x^{3}ln^{5}{10}lg^{2}(x)lg^{2}(lg(x))} + \frac{2}{x^{3}ln^{5}{10}lg^{3}(x)lg(lg(x))} + \frac{3}{x^{3}ln^{6}{10}lg^{3}(x)lg^{2}(lg(x))} + \frac{2}{x^{3}ln^{7}{10}lg^{3}(x)lg^{3}(lg(x))}\right)}{dx}\\=&\frac{2*-3}{x^{4}ln^{3}{10}lg(x)lg(lg(x))} + \frac{2*-3*0}{x^{3}ln^{4}{10}lg(x)lg(lg(x))} + \frac{2*-1}{x^{3}ln^{3}{10}lg^{2}(x)ln{10}(x)lg(lg(x))} + \frac{2*-1}{x^{3}ln^{3}{10}lg(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{4}{10}lg^{2}(x)lg(lg(x))} + \frac{3*-4*0}{x^{3}ln^{5}{10}lg^{2}(x)lg(lg(x))} + \frac{3*-2}{x^{3}ln^{4}{10}lg^{3}(x)ln{10}(x)lg(lg(x))} + \frac{3*-1}{x^{3}ln^{4}{10}lg^{2}(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{5}{10}lg^{2}(x)lg^{2}(lg(x))} + \frac{3*-5*0}{x^{3}ln^{6}{10}lg^{2}(x)lg^{2}(lg(x))} + \frac{3*-2}{x^{3}ln^{5}{10}lg^{3}(x)ln{10}(x)lg^{2}(lg(x))} + \frac{3*-2}{x^{3}ln^{5}{10}lg^{2}(x)lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{5}{10}lg^{3}(x)lg(lg(x))} + \frac{2*-5*0}{x^{3}ln^{6}{10}lg^{3}(x)lg(lg(x))} + \frac{2*-3}{x^{3}ln^{5}{10}lg^{4}(x)ln{10}(x)lg(lg(x))} + \frac{2*-1}{x^{3}ln^{5}{10}lg^{3}(x)lg^{2}(lg(x))ln{10}(lg(x))ln{10}(x)} + \frac{3*-3}{x^{4}ln^{6}{10}lg^{3}(x)lg^{2}(lg(x))} + \frac{3*-6*0}{x^{3}ln^{7}{10}lg^{3}(x)lg^{2}(lg(x))} + \frac{3*-3}{x^{3}ln^{6}{10}lg^{4}(x)ln{10}(x)lg^{2}(lg(x))} + \frac{3*-2}{x^{3}ln^{6}{10}lg^{3}(x)lg^{3}(lg(x))ln{10}(lg(x))ln{10}(x)} + \frac{2*-3}{x^{4}ln^{7}{10}lg^{3}(x)lg^{3}(lg(x))} + \frac{2*-7*0}{x^{3}ln^{8}{10}lg^{3}(x)lg^{3}(lg(x))} + \frac{2*-3}{x^{3}ln^{7}{10}lg^{4}(x)ln{10}(x)lg^{3}(lg(x))} + \frac{2*-3}{x^{3}ln^{7}{10}lg^{3}(x)lg^{4}(lg(x))ln{10}(lg(x))ln{10}(x)}\\=&\frac{-6}{x^{4}ln^{3}{10}lg(x)lg(lg(x))} - \frac{11}{x^{4}ln^{4}{10}lg^{2}(x)lg(lg(x))} - \frac{11}{x^{4}ln^{5}{10}lg^{2}(x)lg^{2}(lg(x))} - \frac{12}{x^{4}ln^{5}{10}lg^{3}(x)lg(lg(x))} - \frac{18}{x^{4}ln^{6}{10}lg^{3}(x)lg^{2}(lg(x))} - \frac{12}{x^{4}ln^{7}{10}lg^{3}(x)lg^{3}(lg(x))} - \frac{6}{x^{4}ln^{6}{10}lg^{4}(x)lg(lg(x))} - \frac{11}{x^{4}ln^{7}{10}lg^{4}(x)lg^{2}(lg(x))} - \frac{12}{x^{4}ln^{8}{10}lg^{4}(x)lg^{3}(lg(x))} - \frac{6}{x^{4}ln^{9}{10}lg^{4}(x)lg^{4}(lg(x))}\\ \end{split}\end{equation} \]

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