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

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