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

\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ lg(x)lg(x) - lg(lg(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg^{2}(x) - lg(lg(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg^{2}(x) - lg(lg(x))\right)}{dx}\\=&\frac{2lg(x)}{ln{10}(x)} - \frac{1}{ln{10}(lg(x))ln{10}(x)}\\=&\frac{2lg(x)}{xln{10}} - \frac{1}{xln^{2}{10}lg(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2lg(x)}{xln{10}} - \frac{1}{xln^{2}{10}lg(x)}\right)}{dx}\\=&\frac{2*-lg(x)}{x^{2}ln{10}} + \frac{2*-0lg(x)}{xln^{2}{10}} + \frac{2}{xln{10}ln{10}(x)} - \frac{-1}{x^{2}ln^{2}{10}lg(x)} - \frac{-2*0}{xln^{3}{10}lg(x)} - \frac{-1}{xln^{2}{10}lg^{2}(x)ln{10}(x)}\\=&\frac{-2lg(x)}{x^{2}ln{10}} + \frac{1}{x^{2}ln^{2}{10}lg(x)} + \frac{1}{x^{2}ln^{3}{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2lg(x)}{x^{2}ln{10}} + \frac{1}{x^{2}ln^{2}{10}lg(x)} + \frac{1}{x^{2}ln^{3}{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}}\right)}{dx}\\=&\frac{-2*-2lg(x)}{x^{3}ln{10}} - \frac{2*-0lg(x)}{x^{2}ln^{2}{10}} - \frac{2}{x^{2}ln{10}ln{10}(x)} + \frac{-2}{x^{3}ln^{2}{10}lg(x)} + \frac{-2*0}{x^{2}ln^{3}{10}lg(x)} + \frac{-1}{x^{2}ln^{2}{10}lg^{2}(x)ln{10}(x)} + \frac{-2}{x^{3}ln^{3}{10}lg^{2}(x)} + \frac{-3*0}{x^{2}ln^{4}{10}lg^{2}(x)} + \frac{-2}{x^{2}ln^{3}{10}lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}{10}} + \frac{2*-2*0}{x^{2}ln^{3}{10}}\\=&\frac{4lg(x)}{x^{3}ln{10}} - \frac{2}{x^{3}ln^{2}{10}lg(x)} - \frac{3}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{2}{x^{3}ln^{4}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{2}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{4lg(x)}{x^{3}ln{10}} - \frac{2}{x^{3}ln^{2}{10}lg(x)} - \frac{3}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{2}{x^{3}ln^{4}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{2}{10}}\right)}{dx}\\=&\frac{4*-3lg(x)}{x^{4}ln{10}} + \frac{4*-0lg(x)}{x^{3}ln^{2}{10}} + \frac{4}{x^{3}ln{10}ln{10}(x)} - \frac{2*-3}{x^{4}ln^{2}{10}lg(x)} - \frac{2*-2*0}{x^{3}ln^{3}{10}lg(x)} - \frac{2*-1}{x^{3}ln^{2}{10}lg^{2}(x)ln{10}(x)} - \frac{3*-3}{x^{4}ln^{3}{10}lg^{2}(x)} - \frac{3*-3*0}{x^{3}ln^{4}{10}lg^{2}(x)} - \frac{3*-2}{x^{3}ln^{3}{10}lg^{3}(x)ln{10}(x)} - \frac{2*-3}{x^{4}ln^{4}{10}lg^{3}(x)} - \frac{2*-4*0}{x^{3}ln^{5}{10}lg^{3}(x)} - \frac{2*-3}{x^{3}ln^{4}{10}lg^{4}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{2}{10}} - \frac{6*-2*0}{x^{3}ln^{3}{10}}\\=&\frac{-12lg(x)}{x^{4}ln{10}} + \frac{6}{x^{4}ln^{2}{10}lg(x)} + \frac{11}{x^{4}ln^{3}{10}lg^{2}(x)} + \frac{12}{x^{4}ln^{4}{10}lg^{3}(x)} + \frac{6}{x^{4}ln^{5}{10}lg^{4}(x)} + \frac{22}{x^{4}ln^{2}{10}}\\ \end{split}\end{equation} \]

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