There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ th(lg(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( th(lg(x))\right)}{dx}\\=&\frac{(1 - th^{2}(lg(x)))}{ln{10}(x)}\\=& - \frac{th^{2}(lg(x))}{xln{10}} + \frac{1}{xln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{th^{2}(lg(x))}{xln{10}} + \frac{1}{xln{10}}\right)}{dx}\\=& - \frac{-th^{2}(lg(x))}{x^{2}ln{10}} - \frac{-0th^{2}(lg(x))}{xln^{2}{10}} - \frac{2th(lg(x))(1 - th^{2}(lg(x)))}{xln{10}ln{10}(x)} + \frac{-1}{x^{2}ln{10}} + \frac{-0}{xln^{2}{10}}\\=&\frac{th^{2}(lg(x))}{x^{2}ln{10}} - \frac{2th(lg(x))}{x^{2}ln^{2}{10}} + \frac{2th^{3}(lg(x))}{x^{2}ln^{2}{10}} - \frac{1}{x^{2}ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{th^{2}(lg(x))}{x^{2}ln{10}} - \frac{2th(lg(x))}{x^{2}ln^{2}{10}} + \frac{2th^{3}(lg(x))}{x^{2}ln^{2}{10}} - \frac{1}{x^{2}ln{10}}\right)}{dx}\\=&\frac{-2th^{2}(lg(x))}{x^{3}ln{10}} + \frac{-0th^{2}(lg(x))}{x^{2}ln^{2}{10}} + \frac{2th(lg(x))(1 - th^{2}(lg(x)))}{x^{2}ln{10}ln{10}(x)} - \frac{2*-2th(lg(x))}{x^{3}ln^{2}{10}} - \frac{2*-2*0th(lg(x))}{x^{2}ln^{3}{10}} - \frac{2(1 - th^{2}(lg(x)))}{x^{2}ln^{2}{10}ln{10}(x)} + \frac{2*-2th^{3}(lg(x))}{x^{3}ln^{2}{10}} + \frac{2*-2*0th^{3}(lg(x))}{x^{2}ln^{3}{10}} + \frac{2*3th^{2}(lg(x))(1 - th^{2}(lg(x)))}{x^{2}ln^{2}{10}ln{10}(x)} - \frac{-2}{x^{3}ln{10}} - \frac{-0}{x^{2}ln^{2}{10}}\\=& - \frac{2th^{2}(lg(x))}{x^{3}ln{10}} + \frac{6th(lg(x))}{x^{3}ln^{2}{10}} - \frac{6th^{3}(lg(x))}{x^{3}ln^{2}{10}} + \frac{8th^{2}(lg(x))}{x^{3}ln^{3}{10}} - \frac{6th^{4}(lg(x))}{x^{3}ln^{3}{10}} - \frac{2}{x^{3}ln^{3}{10}} + \frac{2}{x^{3}ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2th^{2}(lg(x))}{x^{3}ln{10}} + \frac{6th(lg(x))}{x^{3}ln^{2}{10}} - \frac{6th^{3}(lg(x))}{x^{3}ln^{2}{10}} + \frac{8th^{2}(lg(x))}{x^{3}ln^{3}{10}} - \frac{6th^{4}(lg(x))}{x^{3}ln^{3}{10}} - \frac{2}{x^{3}ln^{3}{10}} + \frac{2}{x^{3}ln{10}}\right)}{dx}\\=& - \frac{2*-3th^{2}(lg(x))}{x^{4}ln{10}} - \frac{2*-0th^{2}(lg(x))}{x^{3}ln^{2}{10}} - \frac{2*2th(lg(x))(1 - th^{2}(lg(x)))}{x^{3}ln{10}ln{10}(x)} + \frac{6*-3th(lg(x))}{x^{4}ln^{2}{10}} + \frac{6*-2*0th(lg(x))}{x^{3}ln^{3}{10}} + \frac{6(1 - th^{2}(lg(x)))}{x^{3}ln^{2}{10}ln{10}(x)} - \frac{6*-3th^{3}(lg(x))}{x^{4}ln^{2}{10}} - \frac{6*-2*0th^{3}(lg(x))}{x^{3}ln^{3}{10}} - \frac{6*3th^{2}(lg(x))(1 - th^{2}(lg(x)))}{x^{3}ln^{2}{10}ln{10}(x)} + \frac{8*-3th^{2}(lg(x))}{x^{4}ln^{3}{10}} + \frac{8*-3*0th^{2}(lg(x))}{x^{3}ln^{4}{10}} + \frac{8*2th(lg(x))(1 - th^{2}(lg(x)))}{x^{3}ln^{3}{10}ln{10}(x)} - \frac{6*-3th^{4}(lg(x))}{x^{4}ln^{3}{10}} - \frac{6*-3*0th^{4}(lg(x))}{x^{3}ln^{4}{10}} - \frac{6*4th^{3}(lg(x))(1 - th^{2}(lg(x)))}{x^{3}ln^{3}{10}ln{10}(x)} - \frac{2*-3}{x^{4}ln^{3}{10}} - \frac{2*-3*0}{x^{3}ln^{4}{10}} + \frac{2*-3}{x^{4}ln{10}} + \frac{2*-0}{x^{3}ln^{2}{10}}\\=&\frac{6th^{2}(lg(x))}{x^{4}ln{10}} - \frac{22th(lg(x))}{x^{4}ln^{2}{10}} + \frac{22th^{3}(lg(x))}{x^{4}ln^{2}{10}} - \frac{48th^{2}(lg(x))}{x^{4}ln^{3}{10}} + \frac{36th^{4}(lg(x))}{x^{4}ln^{3}{10}} + \frac{16th(lg(x))}{x^{4}ln^{4}{10}} - \frac{40th^{3}(lg(x))}{x^{4}ln^{4}{10}} + \frac{24th^{5}(lg(x))}{x^{4}ln^{4}{10}} + \frac{12}{x^{4}ln^{3}{10}} - \frac{6}{x^{4}ln{10}}\\ \end{split}\end{equation} \]

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