本次共计算 1 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数cot(lg(x) - lg(x - 1)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cot(lg(x) - lg(x - 1))\right)}{dx}\\=&-csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})\\=&\frac{-csc^{2}(lg(x) - lg(x - 1))}{xln{10}} + \frac{csc^{2}(lg(x) - lg(x - 1))}{(x - 1)ln{10}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-csc^{2}(lg(x) - lg(x - 1))}{xln{10}} + \frac{csc^{2}(lg(x) - lg(x - 1))}{(x - 1)ln{10}}\right)}{dx}\\=&\frac{--csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln{10}} - \frac{-0csc^{2}(lg(x) - lg(x - 1))}{xln^{2}{10}} - \frac{-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{xln{10}} + \frac{(\frac{-(1 + 0)}{(x - 1)^{2}})csc^{2}(lg(x) - lg(x - 1))}{ln{10}} + \frac{-0csc^{2}(lg(x) - lg(x - 1))}{(x - 1)ln^{2}{10}} + \frac{-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)ln{10}}\\=&\frac{csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln{10}} + \frac{2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln^{2}{10}} - \frac{4cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)xln^{2}{10}} - \frac{csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}ln{10}} + \frac{2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}ln^{2}{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln{10}} + \frac{2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln^{2}{10}} - \frac{4cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)xln^{2}{10}} - \frac{csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}ln{10}} + \frac{2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}ln^{2}{10}}\right)}{dx}\\=&\frac{-2csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln{10}} + \frac{-0csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln^{2}{10}} + \frac{-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{x^{2}ln{10}} + \frac{2*-2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{2}{10}} + \frac{2*-2*0cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln^{3}{10}} + \frac{2*-csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln^{2}{10}} + \frac{2cot(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{x^{2}ln^{2}{10}} - \frac{4(\frac{-(1 + 0)}{(x - 1)^{2}})cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{xln^{2}{10}} - \frac{4*-cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{2}{10}} - \frac{4*-2*0cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)xln^{3}{10}} - \frac{4*-csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{(x - 1)xln^{2}{10}} - \frac{4cot(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)xln^{2}{10}} - \frac{(\frac{-2(1 + 0)}{(x - 1)^{3}})csc^{2}(lg(x) - lg(x - 1))}{ln{10}} - \frac{-0csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}ln^{2}{10}} - \frac{-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{2}ln{10}} + \frac{2(\frac{-2(1 + 0)}{(x - 1)^{3}})cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{ln^{2}{10}} + \frac{2*-2*0cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}ln^{3}{10}} + \frac{2*-csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}ln^{2}{10}} + \frac{2cot(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{2}ln^{2}{10}}\\=&\frac{-2csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln{10}} - \frac{6cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{2}{10}} + \frac{6cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{2}{10}} - \frac{2csc^{4}(lg(x) - lg(x - 1))}{x^{3}ln^{3}{10}} + \frac{6csc^{4}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{3}{10}} - \frac{4cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{3}{10}} + \frac{12cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{3}{10}} + \frac{6cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{2}{10}} - \frac{6csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{3}{10}} - \frac{12cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{3}{10}} + \frac{2csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln{10}} - \frac{6cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{2}{10}} + \frac{2csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{3}{10}} + \frac{4cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{3}{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-2csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln{10}} - \frac{6cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{2}{10}} + \frac{6cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{2}{10}} - \frac{2csc^{4}(lg(x) - lg(x - 1))}{x^{3}ln^{3}{10}} + \frac{6csc^{4}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{3}{10}} - \frac{4cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{3}{10}} + \frac{12cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{3}{10}} + \frac{6cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{2}{10}} - \frac{6csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{3}{10}} - \frac{12cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{3}{10}} + \frac{2csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln{10}} - \frac{6cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{2}{10}} + \frac{2csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{3}{10}} + \frac{4cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{3}{10}}\right)}{dx}\\=&\frac{-2*-3csc^{2}(lg(x) - lg(x - 1))}{x^{4}ln{10}} - \frac{2*-0csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{2}{10}} - \frac{2*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{x^{3}ln{10}} - \frac{6*-3cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{4}ln^{2}{10}} - \frac{6*-2*0cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{3}{10}} - \frac{6*-csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{2}{10}} - \frac{6cot(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{x^{3}ln^{2}{10}} + \frac{6(\frac{-(1 + 0)}{(x - 1)^{2}})cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln^{2}{10}} + \frac{6*-2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{3}ln^{2}{10}} + \frac{6*-2*0cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{3}{10}} + \frac{6*-csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{2}{10}} + \frac{6cot(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)x^{2}ln^{2}{10}} - \frac{2*-3csc^{4}(lg(x) - lg(x - 1))}{x^{4}ln^{3}{10}} - \frac{2*-3*0csc^{4}(lg(x) - lg(x - 1))}{x^{3}ln^{4}{10}} - \frac{2*-4csc^{4}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{x^{3}ln^{3}{10}} + \frac{6(\frac{-(1 + 0)}{(x - 1)^{2}})csc^{4}(lg(x) - lg(x - 1))}{x^{2}ln^{3}{10}} + \frac{6*-2csc^{4}(lg(x) - lg(x - 1))}{(x - 1)x^{3}ln^{3}{10}} + \frac{6*-3*0csc^{4}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{4}{10}} + \frac{6*-4csc^{4}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)x^{2}ln^{3}{10}} - \frac{4*-3cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{4}ln^{3}{10}} - \frac{4*-3*0cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{4}{10}} - \frac{4*-2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{x^{3}ln^{3}{10}} - \frac{4cot^{2}(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{x^{3}ln^{3}{10}} + \frac{12(\frac{-(1 + 0)}{(x - 1)^{2}})cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{2}ln^{3}{10}} + \frac{12*-2cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{3}ln^{3}{10}} + \frac{12*-3*0cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{4}{10}} + \frac{12*-2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{2}ln^{3}{10}} + \frac{12cot^{2}(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)x^{2}ln^{3}{10}} + \frac{6(\frac{-2(1 + 0)}{(x - 1)^{3}})cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{xln^{2}{10}} + \frac{6*-cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}x^{2}ln^{2}{10}} + \frac{6*-2*0cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{3}{10}} + \frac{6*-csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{2}{10}} + \frac{6cot(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{2}xln^{2}{10}} - \frac{6(\frac{-2(1 + 0)}{(x - 1)^{3}})csc^{4}(lg(x) - lg(x - 1))}{xln^{3}{10}} - \frac{6*-csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{2}x^{2}ln^{3}{10}} - \frac{6*-3*0csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{4}{10}} - \frac{6*-4csc^{4}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{2}xln^{3}{10}} - \frac{12(\frac{-2(1 + 0)}{(x - 1)^{3}})cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{xln^{3}{10}} - \frac{12*-cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}x^{2}ln^{3}{10}} - \frac{12*-3*0cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{4}{10}} - \frac{12*-2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}xln^{3}{10}} - \frac{12cot^{2}(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{2}xln^{3}{10}} + \frac{2(\frac{-3(1 + 0)}{(x - 1)^{4}})csc^{2}(lg(x) - lg(x - 1))}{ln{10}} + \frac{2*-0csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{2}{10}} + \frac{2*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{3}ln{10}} - \frac{6(\frac{-3(1 + 0)}{(x - 1)^{4}})cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{ln^{2}{10}} - \frac{6*-2*0cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{3}{10}} - \frac{6*-csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{2}{10}} - \frac{6cot(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{3}ln^{2}{10}} + \frac{2(\frac{-3(1 + 0)}{(x - 1)^{4}})csc^{4}(lg(x) - lg(x - 1))}{ln^{3}{10}} + \frac{2*-3*0csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{4}{10}} + \frac{2*-4csc^{4}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{3}ln^{3}{10}} + \frac{4(\frac{-3(1 + 0)}{(x - 1)^{4}})cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{ln^{3}{10}} + \frac{4*-3*0cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{4}{10}} + \frac{4*-2cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}ln^{3}{10}} + \frac{4cot^{2}(lg(x) - lg(x - 1))*-2csc^{2}(lg(x) - lg(x - 1))cot(lg(x) - lg(x - 1))(\frac{1}{ln{10}(x)} - \frac{(1 + 0)}{ln{10}(x - 1)})}{(x - 1)^{3}ln^{3}{10}}\\=&\frac{6csc^{2}(lg(x) - lg(x - 1))}{x^{4}ln{10}} + \frac{16cot(lg(x) - lg(x - 1))csc^{4}(lg(x) - lg(x - 1))}{x^{4}ln^{4}{10}} - \frac{64cot(lg(x) - lg(x - 1))csc^{4}(lg(x) - lg(x - 1))}{(x - 1)x^{3}ln^{4}{10}} + \frac{22cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{4}ln^{2}{10}} + \frac{12csc^{4}(lg(x) - lg(x - 1))}{x^{4}ln^{3}{10}} - \frac{24csc^{4}(lg(x) - lg(x - 1))}{(x - 1)x^{3}ln^{3}{10}} + \frac{24cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{4}ln^{3}{10}} - \frac{48cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{3}ln^{3}{10}} - \frac{12cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}x^{2}ln^{2}{10}} - \frac{16cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{3}ln^{2}{10}} + \frac{96cot(lg(x) - lg(x - 1))csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{2}x^{2}ln^{4}{10}} + \frac{8cot^{3}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{x^{4}ln^{4}{10}} - \frac{32cot^{3}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)x^{3}ln^{4}{10}} - \frac{64cot(lg(x) - lg(x - 1))csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{3}xln^{4}{10}} + \frac{48cot^{3}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{2}x^{2}ln^{4}{10}} - \frac{16cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}xln^{2}{10}} + \frac{24csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{3}xln^{3}{10}} + \frac{48cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}xln^{3}{10}} - \frac{32cot^{3}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{3}xln^{4}{10}} - \frac{6csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{4}ln{10}} + \frac{16cot(lg(x) - lg(x - 1))csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{4}ln^{4}{10}} + \frac{22cot(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{4}ln^{2}{10}} - \frac{12csc^{4}(lg(x) - lg(x - 1))}{(x - 1)^{4}ln^{3}{10}} - \frac{24cot^{2}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{4}ln^{3}{10}} + \frac{8cot^{3}(lg(x) - lg(x - 1))csc^{2}(lg(x) - lg(x - 1))}{(x - 1)^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]

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