求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
本次共计算 3 个题目：每一题对 x 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/3】求函数ln(x + sqrt(xx + 1)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(x + sqrt(x^{2} + 1))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(x + sqrt(x^{2} + 1))\right)}{dx}\\=&\frac{(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))}\\=&\frac{x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} + \frac{1}{(x + sqrt(x^{2} + 1))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} + \frac{1}{(x + sqrt(x^{2} + 1))}\right)}{dx}\\=&\frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} + 1))} + \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} + (\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})\\=& - \frac{x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{2x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(x + sqrt(x^{2} + 1))^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{2x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(x + sqrt(x^{2} + 1))^{2}}\right)}{dx}\\=& - \frac{(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x^{2}}{(x^{2} + 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}}{(x + sqrt(x^{2} + 1))^{2}} - \frac{2x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{2(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} + 1))^{2}} - \frac{2}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x^{2}}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(x + sqrt(x^{2} + 1))} - \frac{2x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})}{(x + sqrt(x^{2} + 1))} - (\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})\\=&\frac{2x^{3}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} - \frac{2x}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{2}} + \frac{6x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{6x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))} - \frac{x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} - \frac{3}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} + \frac{2}{(x + sqrt(x^{2} + 1))^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2x^{3}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} - \frac{2x}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{2}} + \frac{6x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{6x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))} - \frac{x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} - \frac{3}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} + \frac{2}{(x + sqrt(x^{2} + 1))^{3}}\right)}{dx}\\=&\frac{2(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{4}})x^{3}}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{3}}{(x + sqrt(x^{2} + 1))^{3}} + \frac{2*3x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x^{3}}{(x^{2} + 1)^{2}} + \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}}{(x + sqrt(x^{2} + 1))^{2}} + \frac{3*3x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} - \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(x + sqrt(x^{2} + 1))^{2}} - \frac{2(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x}{(x^{2} + 1)} - \frac{2}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{2}} + \frac{6(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{4}})x^{2}}{(x^{2} + 1)} + \frac{6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}}{(x + sqrt(x^{2} + 1))^{3}} + \frac{6*2x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{6(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{4}})x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{6(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} + 1))^{3}} + \frac{6}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x^{2}}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(x + sqrt(x^{2} + 1))^{2}} + \frac{3*2x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x^{3}}{(x^{2} + 1)^{\frac{5}{2}}} + \frac{3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{3}}{(x + sqrt(x^{2} + 1))} + \frac{3*3x^{2}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x}{(x + sqrt(x^{2} + 1))} - \frac{2(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))} - \frac{(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x}{(x^{2} + 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(x + sqrt(x^{2} + 1))^{2}} - \frac{1}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x}{(x + sqrt(x^{2} + 1))} - \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} - \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{3(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})}{(x + sqrt(x^{2} + 1))^{2}} + 2(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{4}})\\=& - \frac{6x^{4}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{2}} - \frac{6x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} - \frac{12x^{4}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{5}{2}}} + \frac{6x^{2}}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))^{3}} - \frac{24x^{3}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{2}} - \frac{15x^{4}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{3}} + \frac{13x^{2}}{(x^{2} + 1)^{2}(x + sqrt(x^{2} + 1))^{2}} - \frac{24x^{3}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{\frac{3}{2}}} - \frac{36x^{2}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)} + \frac{12x}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{3}} - \frac{24x}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{\frac{1}{2}}} + \frac{5x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} - \frac{12x^{3}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{5}{2}}} + \frac{6x}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))^{2}} - \frac{15x^{4}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{7}{2}}} + \frac{15x^{2}}{(x^{2} + 1)^{\frac{5}{2}}(x + sqrt(x^{2} + 1))} + \frac{6x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{12x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{3x^{2}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} + \frac{12}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{1}{2}}} - \frac{2}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{2}} - \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))} - \frac{6}{(x + sqrt(x^{2} + 1))^{4}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/3】求函数ln(x + sqrt(xx - 1)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(x + sqrt(x^{2} - 1))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(x + sqrt(x^{2} - 1))\right)}{dx}\\=&\frac{(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))}\\=&\frac{x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} + \frac{1}{(x + sqrt(x^{2} - 1))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} + \frac{1}{(x + sqrt(x^{2} - 1))}\right)}{dx}\\=&\frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x}{(x^{2} - 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} - 1))} + \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} + (\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})\\=& - \frac{x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{2x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} + \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} - \frac{1}{(x + sqrt(x^{2} - 1))^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{2x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} + \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} - \frac{1}{(x + sqrt(x^{2} - 1))^{2}}\right)}{dx}\\=& - \frac{(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x^{2}}{(x^{2} - 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})x^{2}}{(x + sqrt(x^{2} - 1))^{2}} - \frac{2x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{2(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x}{(x^{2} - 1)^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} - 1))^{2}} - \frac{2}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} - \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x^{2}}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x^{2}}{(x + sqrt(x^{2} - 1))} - \frac{2x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} + \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})}{(x^{2} - 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})}{(x + sqrt(x^{2} - 1))} - (\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})\\=&\frac{2x^{3}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{2}} - \frac{2x}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{2}} + \frac{6x^{2}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)} + \frac{6x}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))} - \frac{x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} - \frac{3}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} + \frac{2}{(x + sqrt(x^{2} - 1))^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2x^{3}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{2}} - \frac{2x}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{2}} + \frac{6x^{2}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)} + \frac{6x}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))} - \frac{x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} - \frac{3}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} + \frac{2}{(x + sqrt(x^{2} - 1))^{3}}\right)}{dx}\\=&\frac{2(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{4}})x^{3}}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x^{3}}{(x + sqrt(x^{2} - 1))^{3}} + \frac{2*3x^{2}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x^{3}}{(x^{2} - 1)^{2}} + \frac{3(\frac{-2(2x + 0)}{(x^{2} - 1)^{3}})x^{3}}{(x + sqrt(x^{2} - 1))^{2}} + \frac{3*3x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{2}} - \frac{2(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})x}{(x + sqrt(x^{2} - 1))^{2}} - \frac{2(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x}{(x^{2} - 1)} - \frac{2}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{2}} + \frac{6(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{4}})x^{2}}{(x^{2} - 1)} + \frac{6(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})x^{2}}{(x + sqrt(x^{2} - 1))^{3}} + \frac{6*2x}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)} + \frac{6(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{4}})x}{(x^{2} - 1)^{\frac{1}{2}}} + \frac{6(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} - 1))^{3}} + \frac{6}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{1}{2}}} + \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x^{2}}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x^{2}}{(x + sqrt(x^{2} - 1))^{2}} + \frac{3*2x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x^{3}}{(x^{2} - 1)^{\frac{5}{2}}} + \frac{3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} - 1)^{\frac{7}{2}}})x^{3}}{(x + sqrt(x^{2} - 1))} + \frac{3*3x^{2}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{5}{2}}} - \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x}{(x + sqrt(x^{2} - 1))} - \frac{2(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))} - \frac{(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x}{(x^{2} - 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})x}{(x + sqrt(x^{2} - 1))^{2}} - \frac{1}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x}{(x + sqrt(x^{2} - 1))} - \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} - \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})}{(x^{2} - 1)^{\frac{1}{2}}} - \frac{3(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})}{(x + sqrt(x^{2} - 1))^{2}} + 2(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{4}})\\=& - \frac{6x^{4}}{(x + sqrt(x^{2} - 1))^{4}(x^{2} - 1)^{2}} - \frac{6x^{2}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{3}{2}}} - \frac{12x^{4}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{5}{2}}} + \frac{6x^{2}}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))^{3}} - \frac{24x^{3}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{2}} - \frac{15x^{4}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{3}} + \frac{13x^{2}}{(x^{2} - 1)^{2}(x + sqrt(x^{2} - 1))^{2}} - \frac{24x^{3}}{(x + sqrt(x^{2} - 1))^{4}(x^{2} - 1)^{\frac{3}{2}}} - \frac{36x^{2}}{(x + sqrt(x^{2} - 1))^{4}(x^{2} - 1)} + \frac{12x}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{3}} - \frac{24x}{(x + sqrt(x^{2} - 1))^{4}(x^{2} - 1)^{\frac{1}{2}}} + \frac{5x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{2}} - \frac{12x^{3}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{5}{2}}} + \frac{6x}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))^{2}} - \frac{15x^{4}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{7}{2}}} + \frac{15x^{2}}{(x^{2} - 1)^{\frac{5}{2}}(x + sqrt(x^{2} - 1))} + \frac{6x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{12x}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)} + \frac{3x^{2}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{5}{2}}} - \frac{1}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} + \frac{12}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{1}{2}}} - \frac{2}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{2}} - \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))} - \frac{6}{(x + sqrt(x^{2} - 1))^{4}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【3/3】求函数\frac{ln(\frac{(x + 1)}{(x - 1)})}{2} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{1}{2}ln(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{1}{2}ln(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})\right)}{dx}\\=&\frac{\frac{1}{2}((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})}\\=&\frac{-x}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{1}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})(x - 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-x}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{1}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})(x - 1)}\right)}{dx}\\=&\frac{-(\frac{-2(1 + 0)}{(x - 1)^{3}})x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})x}{2(x - 1)^{2}} - \frac{1}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-2(1 + 0)}{(x - 1)^{3}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{2(x - 1)^{2}} + \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{2(x - 1)} + \frac{(\frac{-(1 + 0)}{(x - 1)^{2}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})}\\=&\frac{x}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{x^{2}}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{x}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{1}{(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{1}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{x}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{x^{2}}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{x}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{1}{(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{1}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{2}}\right)}{dx}\\=&\frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})x}{(x - 1)^{3}} + \frac{1}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})x^{2}}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x^{2}}{2(x - 1)^{4}} - \frac{2x}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{2(x - 1)^{3}} + \frac{1}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{(x - 1)^{4}} - \frac{1}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{2(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{(\frac{-2(1 + 0)}{(x - 1)^{3}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{(x - 1)^{2}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{4}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{3}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{2}} - \frac{(\frac{-2(1 + 0)}{(x - 1)^{3}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}}\\=&\frac{-3x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{3x^{2}}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{5x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{4}} - \frac{x^{3}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2x^{2}}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{4x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} + \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{6}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{3}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{1}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} + \frac{5}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{2}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} - \frac{1}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{3}} + \frac{3}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-3x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{3x^{2}}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{5x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{4}} - \frac{x^{3}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2x^{2}}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{4x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} + \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{6}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{3}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{1}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} + \frac{5}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{2}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} - \frac{1}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{3}} + \frac{3}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})}\right)}{dx}\\=&\frac{-3(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{3(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})x}{(x - 1)^{4}} - \frac{3}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{3(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{3(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x^{2}}{(x - 1)^{5}} + \frac{3*2x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{5(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{5(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{(x - 1)^{4}} - \frac{5}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6(\frac{-5(1 + 0)}{(x - 1)^{6}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{(x - 1)^{5}} + \frac{6}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{(x - 1)^{4}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{4}} - \frac{(\frac{-6(1 + 0)}{(x - 1)^{7}})x^{3}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x^{3}}{(x - 1)^{6}} - \frac{3x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x^{2}}{(x - 1)^{5}} + \frac{2*2x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3(\frac{-6(1 + 0)}{(x - 1)^{7}})x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x^{2}}{(x - 1)^{6}} - \frac{3*2x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x^{2}}{(x - 1)^{5}} + \frac{(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{4}} - \frac{1}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{4(\frac{-5(1 + 0)}{(x - 1)^{6}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{4(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{5}} + \frac{4}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3(\frac{-6(1 + 0)}{(x - 1)^{7}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{6}} - \frac{3}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{4}} - \frac{2(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{2}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} + \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{5}} + \frac{2(\frac{-5(1 + 0)}{(x - 1)^{6}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{6(\frac{-4(1 + 0)}{(x - 1)^{5}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{6(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{(x - 1)^{4}} + \frac{3(\frac{-5(1 + 0)}{(x - 1)^{6}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{3(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{(x - 1)^{5}} - \frac{(\frac{-6(1 + 0)}{(x - 1)^{7}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{6}} + \frac{2(\frac{-5(1 + 0)}{(x - 1)^{6}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{5}} + \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{5}} + \frac{(\frac{-5(1 + 0)}{(x - 1)^{6}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{5(\frac{-3(1 + 0)}{(x - 1)^{4}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{5(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{3}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{4}} - \frac{2(\frac{-4(1 + 0)}{(x - 1)^{5}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{4}} - \frac{3(\frac{-4(1 + 0)}{(x - 1)^{5}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{3(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{(x - 1)^{4}} + \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{3(\frac{-3(1 + 0)}{(x - 1)^{4}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{3(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{(x - 1)^{3}}\\=&\frac{12x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{18x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{3x^{4}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}} - \frac{36x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{5}} + \frac{12x^{3}}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{31x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{36x^{2}}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{30x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{7x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{5x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{6}} + \frac{29x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{64x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{9x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} - \frac{9x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{6}} - \frac{8x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{6}} - \frac{12x^{3}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}} + \frac{6x^{3}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{7}} - \frac{18x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}} + \frac{6x^{3}}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{18x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{7}} + \frac{18x^{2}}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{3x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} - \frac{18x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{18x}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{36x}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{18x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{7}} - \frac{12x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}} + \frac{9x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{5}} - \frac{18x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{6}} - \frac{17}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{3}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} + \frac{36}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{33}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{4}} - \frac{9}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{3}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} + \frac{12}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{12}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{6}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{6}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{7}} + \frac{12}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{9}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} - \frac{9}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{6}} - \frac{36}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{3}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{9}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{5}} - \frac{18}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{3}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{4}} - \frac{3}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/3】求函数ln(x + sqrt(xx + 1)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(x + sqrt(x^{2} + 1))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(x + sqrt(x^{2} + 1))\right)}{dx}\\=&\frac{(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))}\\=&\frac{x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} + \frac{1}{(x + sqrt(x^{2} + 1))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} + \frac{1}{(x + sqrt(x^{2} + 1))}\right)}{dx}\\=&\frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} + 1))} + \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} + (\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})\\=& - \frac{x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{2x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(x + sqrt(x^{2} + 1))^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{2x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(x + sqrt(x^{2} + 1))^{2}}\right)}{dx}\\=& - \frac{(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x^{2}}{(x^{2} + 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}}{(x + sqrt(x^{2} + 1))^{2}} - \frac{2x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{2(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} + 1))^{2}} - \frac{2}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x^{2}}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(x + sqrt(x^{2} + 1))} - \frac{2x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})}{(x + sqrt(x^{2} + 1))} - (\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})\\=&\frac{2x^{3}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} - \frac{2x}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{2}} + \frac{6x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{6x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))} - \frac{x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} - \frac{3}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} + \frac{2}{(x + sqrt(x^{2} + 1))^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2x^{3}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} - \frac{2x}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{2}} + \frac{6x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{6x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))} - \frac{x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{x}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} - \frac{3}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{1}{2}}} + \frac{2}{(x + sqrt(x^{2} + 1))^{3}}\right)}{dx}\\=&\frac{2(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{4}})x^{3}}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{3}}{(x + sqrt(x^{2} + 1))^{3}} + \frac{2*3x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x^{3}}{(x^{2} + 1)^{2}} + \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}}{(x + sqrt(x^{2} + 1))^{2}} + \frac{3*3x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} - \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(x + sqrt(x^{2} + 1))^{2}} - \frac{2(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x}{(x^{2} + 1)} - \frac{2}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{2}} + \frac{6(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{4}})x^{2}}{(x^{2} + 1)} + \frac{6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}}{(x + sqrt(x^{2} + 1))^{3}} + \frac{6*2x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{6(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{4}})x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{6(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} + 1))^{3}} + \frac{6}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x^{2}}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(x + sqrt(x^{2} + 1))^{2}} + \frac{3*2x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x^{3}}{(x^{2} + 1)^{\frac{5}{2}}} + \frac{3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{3}}{(x + sqrt(x^{2} + 1))} + \frac{3*3x^{2}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x}{(x + sqrt(x^{2} + 1))} - \frac{2(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))} - \frac{(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})x}{(x^{2} + 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(x + sqrt(x^{2} + 1))^{2}} - \frac{1}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} - \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{2}})x}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x}{(x + sqrt(x^{2} + 1))} - \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} - \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{3}})}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{3(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})}{(x + sqrt(x^{2} + 1))^{2}} + 2(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} + 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} + 1))^{4}})\\=& - \frac{6x^{4}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{2}} - \frac{6x^{2}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{3}{2}}} - \frac{12x^{4}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{5}{2}}} + \frac{6x^{2}}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))^{3}} - \frac{24x^{3}}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{2}} - \frac{15x^{4}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{3}} + \frac{13x^{2}}{(x^{2} + 1)^{2}(x + sqrt(x^{2} + 1))^{2}} - \frac{24x^{3}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{\frac{3}{2}}} - \frac{36x^{2}}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)} + \frac{12x}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{3}} - \frac{24x}{(x + sqrt(x^{2} + 1))^{4}(x^{2} + 1)^{\frac{1}{2}}} + \frac{5x^{2}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{2}} - \frac{12x^{3}}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{5}{2}}} + \frac{6x}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))^{2}} - \frac{15x^{4}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{7}{2}}} + \frac{15x^{2}}{(x^{2} + 1)^{\frac{5}{2}}(x + sqrt(x^{2} + 1))} + \frac{6x}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{12x}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)} + \frac{3x^{2}}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(x + sqrt(x^{2} + 1))^{2}(x^{2} + 1)} + \frac{12}{(x + sqrt(x^{2} + 1))^{3}(x^{2} + 1)^{\frac{1}{2}}} - \frac{2}{(x^{2} + 1)(x + sqrt(x^{2} + 1))^{2}} - \frac{1}{(x + sqrt(x^{2} + 1))(x^{2} + 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} + 1)^{\frac{3}{2}}(x + sqrt(x^{2} + 1))} - \frac{6}{(x + sqrt(x^{2} + 1))^{4}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/3】求函数ln(x + sqrt(xx - 1)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(x + sqrt(x^{2} - 1))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(x + sqrt(x^{2} - 1))\right)}{dx}\\=&\frac{(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))}\\=&\frac{x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} + \frac{1}{(x + sqrt(x^{2} - 1))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} + \frac{1}{(x + sqrt(x^{2} - 1))}\right)}{dx}\\=&\frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x}{(x^{2} - 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} - 1))} + \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} + (\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})\\=& - \frac{x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{2x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} + \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} - \frac{1}{(x + sqrt(x^{2} - 1))^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{2x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} + \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{1}{2}}} - \frac{1}{(x + sqrt(x^{2} - 1))^{2}}\right)}{dx}\\=& - \frac{(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x^{2}}{(x^{2} - 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})x^{2}}{(x + sqrt(x^{2} - 1))^{2}} - \frac{2x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{2(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x}{(x^{2} - 1)^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} - 1))^{2}} - \frac{2}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} - \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x^{2}}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x^{2}}{(x + sqrt(x^{2} - 1))} - \frac{2x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} + \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})}{(x^{2} - 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})}{(x + sqrt(x^{2} - 1))} - (\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})\\=&\frac{2x^{3}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{2}} - \frac{2x}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{2}} + \frac{6x^{2}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)} + \frac{6x}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))} - \frac{x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} - \frac{3}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} + \frac{2}{(x + sqrt(x^{2} - 1))^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2x^{3}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{2}} - \frac{2x}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{2}} + \frac{6x^{2}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)} + \frac{6x}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))} - \frac{x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{x}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} - \frac{3}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{1}{2}}} + \frac{2}{(x + sqrt(x^{2} - 1))^{3}}\right)}{dx}\\=&\frac{2(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{4}})x^{3}}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x^{3}}{(x + sqrt(x^{2} - 1))^{3}} + \frac{2*3x^{2}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x^{3}}{(x^{2} - 1)^{2}} + \frac{3(\frac{-2(2x + 0)}{(x^{2} - 1)^{3}})x^{3}}{(x + sqrt(x^{2} - 1))^{2}} + \frac{3*3x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{2}} - \frac{2(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})x}{(x + sqrt(x^{2} - 1))^{2}} - \frac{2(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x}{(x^{2} - 1)} - \frac{2}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{2}} + \frac{6(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{4}})x^{2}}{(x^{2} - 1)} + \frac{6(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})x^{2}}{(x + sqrt(x^{2} - 1))^{3}} + \frac{6*2x}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)} + \frac{6(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{4}})x}{(x^{2} - 1)^{\frac{1}{2}}} + \frac{6(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x}{(x + sqrt(x^{2} - 1))^{3}} + \frac{6}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{1}{2}}} + \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x^{2}}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x^{2}}{(x + sqrt(x^{2} - 1))^{2}} + \frac{3*2x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{3(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x^{3}}{(x^{2} - 1)^{\frac{5}{2}}} + \frac{3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} - 1)^{\frac{7}{2}}})x^{3}}{(x + sqrt(x^{2} - 1))} + \frac{3*3x^{2}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{5}{2}}} - \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x}{(x + sqrt(x^{2} - 1))} - \frac{2(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))} - \frac{(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})x}{(x^{2} - 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} - 1)^{2}})x}{(x + sqrt(x^{2} - 1))^{2}} - \frac{1}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} - \frac{(\frac{-(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{2}})x}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x}{(x + sqrt(x^{2} - 1))} - \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} - \frac{3(\frac{-2(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{3}})}{(x^{2} - 1)^{\frac{1}{2}}} - \frac{3(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})}{(x + sqrt(x^{2} - 1))^{2}} + 2(\frac{-3(1 + \frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(x + sqrt(x^{2} - 1))^{4}})\\=& - \frac{6x^{4}}{(x + sqrt(x^{2} - 1))^{4}(x^{2} - 1)^{2}} - \frac{6x^{2}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{3}{2}}} - \frac{12x^{4}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{5}{2}}} + \frac{6x^{2}}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))^{3}} - \frac{24x^{3}}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{2}} - \frac{15x^{4}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{3}} + \frac{13x^{2}}{(x^{2} - 1)^{2}(x + sqrt(x^{2} - 1))^{2}} - \frac{24x^{3}}{(x + sqrt(x^{2} - 1))^{4}(x^{2} - 1)^{\frac{3}{2}}} - \frac{36x^{2}}{(x + sqrt(x^{2} - 1))^{4}(x^{2} - 1)} + \frac{12x}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{3}} - \frac{24x}{(x + sqrt(x^{2} - 1))^{4}(x^{2} - 1)^{\frac{1}{2}}} + \frac{5x^{2}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{2}} - \frac{12x^{3}}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{5}{2}}} + \frac{6x}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))^{2}} - \frac{15x^{4}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{7}{2}}} + \frac{15x^{2}}{(x^{2} - 1)^{\frac{5}{2}}(x + sqrt(x^{2} - 1))} + \frac{6x}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{12x}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)} + \frac{3x^{2}}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{5}{2}}} - \frac{1}{(x + sqrt(x^{2} - 1))^{2}(x^{2} - 1)} + \frac{12}{(x + sqrt(x^{2} - 1))^{3}(x^{2} - 1)^{\frac{1}{2}}} - \frac{2}{(x^{2} - 1)(x + sqrt(x^{2} - 1))^{2}} - \frac{1}{(x + sqrt(x^{2} - 1))(x^{2} - 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} - 1)^{\frac{3}{2}}(x + sqrt(x^{2} - 1))} - \frac{6}{(x + sqrt(x^{2} - 1))^{4}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【3/3】求函数\frac{ln(\frac{(x + 1)}{(x - 1)})}{2} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{1}{2}ln(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{1}{2}ln(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})\right)}{dx}\\=&\frac{\frac{1}{2}((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})}\\=&\frac{-x}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{1}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})(x - 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-x}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{1}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})(x - 1)}\right)}{dx}\\=&\frac{-(\frac{-2(1 + 0)}{(x - 1)^{3}})x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})x}{2(x - 1)^{2}} - \frac{1}{2(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-2(1 + 0)}{(x - 1)^{3}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{2(x - 1)^{2}} + \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{2(x - 1)} + \frac{(\frac{-(1 + 0)}{(x - 1)^{2}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})}\\=&\frac{x}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{x^{2}}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{x}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{1}{(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{1}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{x}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{x^{2}}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{x}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{1}{(x - 1)^{2}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{1}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{2}}\right)}{dx}\\=&\frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})x}{(x - 1)^{3}} + \frac{1}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})x^{2}}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x^{2}}{2(x - 1)^{4}} - \frac{2x}{2(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{2(x - 1)^{3}} + \frac{1}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{(x - 1)^{4}} - \frac{1}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{2(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})x}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{(\frac{-2(1 + 0)}{(x - 1)^{3}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{(x - 1)^{2}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{4}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{3}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{2}} - \frac{(\frac{-2(1 + 0)}{(x - 1)^{3}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}}\\=&\frac{-3x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{3x^{2}}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{5x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{4}} - \frac{x^{3}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2x^{2}}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{4x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} + \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{6}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{3}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{1}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} + \frac{5}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{2}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} - \frac{1}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{3}} + \frac{3}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-3x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{3x^{2}}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{5x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{4}} - \frac{x^{3}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2x^{2}}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{x}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{4x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} + \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{6}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{3}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{1}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} + \frac{5}{2(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{1}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{3}} - \frac{2}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} - \frac{1}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{3}} + \frac{3}{(x - 1)^{3}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})}\right)}{dx}\\=&\frac{-3(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{3(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})x}{(x - 1)^{4}} - \frac{3}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{3(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{3(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x^{2}}{(x - 1)^{5}} + \frac{3*2x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{5(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{5(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{(x - 1)^{4}} - \frac{5}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6(\frac{-5(1 + 0)}{(x - 1)^{6}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{(x - 1)^{5}} + \frac{6}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})x}{(x - 1)^{4}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{4}} - \frac{(\frac{-6(1 + 0)}{(x - 1)^{7}})x^{3}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x^{3}}{(x - 1)^{6}} - \frac{3x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x^{2}}{(x - 1)^{5}} + \frac{2*2x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3(\frac{-6(1 + 0)}{(x - 1)^{7}})x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x^{2}}{(x - 1)^{6}} - \frac{3*2x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x^{2}}{(x - 1)^{5}} + \frac{(\frac{-5(1 + 0)}{(x - 1)^{6}})x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{4}} - \frac{1}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{4(\frac{-5(1 + 0)}{(x - 1)^{6}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{4(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{5}} + \frac{4}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3(\frac{-6(1 + 0)}{(x - 1)^{7}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{3(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{6}} - \frac{3}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{4}} - \frac{2(\frac{-4(1 + 0)}{(x - 1)^{5}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{2}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} + \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})x}{(x - 1)^{5}} + \frac{2(\frac{-5(1 + 0)}{(x - 1)^{6}})x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{6(\frac{-4(1 + 0)}{(x - 1)^{5}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{6(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{(x - 1)^{4}} + \frac{3(\frac{-5(1 + 0)}{(x - 1)^{6}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{3(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{(x - 1)^{5}} - \frac{(\frac{-6(1 + 0)}{(x - 1)^{7}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{6}} + \frac{2(\frac{-5(1 + 0)}{(x - 1)^{6}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{5}} + \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{5}} + \frac{(\frac{-5(1 + 0)}{(x - 1)^{6}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{5(\frac{-3(1 + 0)}{(x - 1)^{4}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{5(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{3}} + \frac{(\frac{-2((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}})}{2(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{2(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{2(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{4}} - \frac{2(\frac{-4(1 + 0)}{(x - 1)^{5}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-4(1 + 0)}{(x - 1)^{5}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{4}} - \frac{3(\frac{-4(1 + 0)}{(x - 1)^{5}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{3(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{(x - 1)^{4}} + \frac{(\frac{-3((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}})}{(x - 1)^{3}} + \frac{(\frac{-3(1 + 0)}{(x - 1)^{4}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{3(\frac{-3(1 + 0)}{(x - 1)^{4}})}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{3(\frac{-((\frac{-(1 + 0)}{(x - 1)^{2}})x + \frac{1}{(x - 1)} + (\frac{-(1 + 0)}{(x - 1)^{2}}))}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}})}{(x - 1)^{3}}\\=&\frac{12x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{18x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{3x^{4}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}} - \frac{36x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{6x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{5}} + \frac{12x^{3}}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{31x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{36x^{2}}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{30x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{7x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} - \frac{5x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{6}} + \frac{29x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{64x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{9x^{2}}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} - \frac{9x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{6}} - \frac{8x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{6}} - \frac{12x^{3}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}} + \frac{6x^{3}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{7}} - \frac{18x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}} + \frac{6x^{3}}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{18x^{2}}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{7}} + \frac{18x^{2}}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{3x}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} - \frac{18x}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{18x}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{36x}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{18x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{7}} - \frac{12x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}} + \frac{9x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{5}} - \frac{18x}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{6}} - \frac{17}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{3}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{4}} + \frac{36}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} + \frac{33}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{1}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}(x - 1)^{4}} - \frac{9}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{3}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}(x - 1)^{5}} + \frac{12}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} - \frac{12}{(x - 1)^{4}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})} + \frac{6}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{6}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{7}} + \frac{12}{(x - 1)^{7}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} - \frac{9}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} - \frac{9}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{6}} - \frac{36}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{3}} + \frac{3}{(x - 1)^{5}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}} + \frac{9}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{5}} - \frac{18}{(x - 1)^{6}(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{2}} - \frac{3}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{4}} - \frac{3}{(\frac{x}{(x - 1)} + \frac{1}{(x - 1)})^{4}(x - 1)^{8}}\\ \end{split}\end{equation} \]
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