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

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