There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {lg({lg(x)}^{2})}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = lg^{2}(lg^{2}(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg^{2}(lg^{2}(x))\right)}{dx}\\=&\frac{2lg(lg^{2}(x))*2lg(x)}{ln{10}(lg^{2}(x))ln{10}(x)}\\=&\frac{4lg(lg^{2}(x))}{xln^{2}{10}lg(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{4lg(lg^{2}(x))}{xln^{2}{10}lg(x)}\right)}{dx}\\=&\frac{4*-lg(lg^{2}(x))}{x^{2}ln^{2}{10}lg(x)} + \frac{4*-2*0lg(lg^{2}(x))}{xln^{3}{10}lg(x)} + \frac{4*-lg(lg^{2}(x))}{xln^{2}{10}lg^{2}(x)ln{10}(x)} + \frac{4*2lg(x)}{xln^{2}{10}lg(x)ln{10}(lg^{2}(x))ln{10}(x)}\\=&\frac{-4lg(lg^{2}(x))}{x^{2}ln^{2}{10}lg(x)} - \frac{4lg(lg^{2}(x))}{x^{2}ln^{3}{10}lg^{2}(x)} + \frac{8}{x^{2}ln^{4}{10}lg^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4lg(lg^{2}(x))}{x^{2}ln^{2}{10}lg(x)} - \frac{4lg(lg^{2}(x))}{x^{2}ln^{3}{10}lg^{2}(x)} + \frac{8}{x^{2}ln^{4}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{-4*-2lg(lg^{2}(x))}{x^{3}ln^{2}{10}lg(x)} - \frac{4*-2*0lg(lg^{2}(x))}{x^{2}ln^{3}{10}lg(x)} - \frac{4*-lg(lg^{2}(x))}{x^{2}ln^{2}{10}lg^{2}(x)ln{10}(x)} - \frac{4*2lg(x)}{x^{2}ln^{2}{10}lg(x)ln{10}(lg^{2}(x))ln{10}(x)} - \frac{4*-2lg(lg^{2}(x))}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{4*-3*0lg(lg^{2}(x))}{x^{2}ln^{4}{10}lg^{2}(x)} - \frac{4*-2lg(lg^{2}(x))}{x^{2}ln^{3}{10}lg^{3}(x)ln{10}(x)} - \frac{4*2lg(x)}{x^{2}ln^{3}{10}lg^{2}(x)ln{10}(lg^{2}(x))ln{10}(x)} + \frac{8*-2}{x^{3}ln^{4}{10}lg^{2}(x)} + \frac{8*-4*0}{x^{2}ln^{5}{10}lg^{2}(x)} + \frac{8*-2}{x^{2}ln^{4}{10}lg^{3}(x)ln{10}(x)}\\=&\frac{8lg(lg^{2}(x))}{x^{3}ln^{2}{10}lg(x)} + \frac{12lg(lg^{2}(x))}{x^{3}ln^{3}{10}lg^{2}(x)} + \frac{8lg(lg^{2}(x))}{x^{3}ln^{4}{10}lg^{3}(x)} - \frac{24}{x^{3}ln^{4}{10}lg^{2}(x)} - \frac{24}{x^{3}ln^{5}{10}lg^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8lg(lg^{2}(x))}{x^{3}ln^{2}{10}lg(x)} + \frac{12lg(lg^{2}(x))}{x^{3}ln^{3}{10}lg^{2}(x)} + \frac{8lg(lg^{2}(x))}{x^{3}ln^{4}{10}lg^{3}(x)} - \frac{24}{x^{3}ln^{4}{10}lg^{2}(x)} - \frac{24}{x^{3}ln^{5}{10}lg^{3}(x)}\right)}{dx}\\=&\frac{8*-3lg(lg^{2}(x))}{x^{4}ln^{2}{10}lg(x)} + \frac{8*-2*0lg(lg^{2}(x))}{x^{3}ln^{3}{10}lg(x)} + \frac{8*-lg(lg^{2}(x))}{x^{3}ln^{2}{10}lg^{2}(x)ln{10}(x)} + \frac{8*2lg(x)}{x^{3}ln^{2}{10}lg(x)ln{10}(lg^{2}(x))ln{10}(x)} + \frac{12*-3lg(lg^{2}(x))}{x^{4}ln^{3}{10}lg^{2}(x)} + \frac{12*-3*0lg(lg^{2}(x))}{x^{3}ln^{4}{10}lg^{2}(x)} + \frac{12*-2lg(lg^{2}(x))}{x^{3}ln^{3}{10}lg^{3}(x)ln{10}(x)} + \frac{12*2lg(x)}{x^{3}ln^{3}{10}lg^{2}(x)ln{10}(lg^{2}(x))ln{10}(x)} + \frac{8*-3lg(lg^{2}(x))}{x^{4}ln^{4}{10}lg^{3}(x)} + \frac{8*-4*0lg(lg^{2}(x))}{x^{3}ln^{5}{10}lg^{3}(x)} + \frac{8*-3lg(lg^{2}(x))}{x^{3}ln^{4}{10}lg^{4}(x)ln{10}(x)} + \frac{8*2lg(x)}{x^{3}ln^{4}{10}lg^{3}(x)ln{10}(lg^{2}(x))ln{10}(x)} - \frac{24*-3}{x^{4}ln^{4}{10}lg^{2}(x)} - \frac{24*-4*0}{x^{3}ln^{5}{10}lg^{2}(x)} - \frac{24*-2}{x^{3}ln^{4}{10}lg^{3}(x)ln{10}(x)} - \frac{24*-3}{x^{4}ln^{5}{10}lg^{3}(x)} - \frac{24*-5*0}{x^{3}ln^{6}{10}lg^{3}(x)} - \frac{24*-3}{x^{3}ln^{5}{10}lg^{4}(x)ln{10}(x)}\\=&\frac{-24lg(lg^{2}(x))}{x^{4}ln^{2}{10}lg(x)} - \frac{44lg(lg^{2}(x))}{x^{4}ln^{3}{10}lg^{2}(x)} - \frac{48lg(lg^{2}(x))}{x^{4}ln^{4}{10}lg^{3}(x)} - \frac{24lg(lg^{2}(x))}{x^{4}ln^{5}{10}lg^{4}(x)} + \frac{144}{x^{4}ln^{5}{10}lg^{3}(x)} + \frac{88}{x^{4}ln^{4}{10}lg^{2}(x)} + \frac{88}{x^{4}ln^{6}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]

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