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

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