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。
\[ \begin{equation}\begin{split}【1/1】求函数ln(\frac{2}{(tan(\frac{x}{2}) - 1)} + 1) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)\right)}{dx}\\=&\frac{(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)}\\=&\frac{-sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\right)}{dx}\\=&\frac{-(\frac{-(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}})sec^{2}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{2}} - \frac{(\frac{-2(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{3}})sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{2sec^{2}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\\=&\frac{-sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{4}} + \frac{sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{3}} - \frac{tan(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{4}} + \frac{sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{3}} - \frac{tan(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\right)}{dx}\\=&\frac{-(\frac{-2(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}})sec^{4}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{4}} - \frac{(\frac{-4(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{5}})sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}} - \frac{4sec^{4}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{4}} + \frac{(\frac{-(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}})sec^{4}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{3}} + \frac{(\frac{-3(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{4}})sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} + \frac{4sec^{4}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{3}} - \frac{(\frac{-(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}})tan(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{2}} - \frac{(\frac{-2(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{3}})tan(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{sec^{2}(\frac{1}{2}x)(\frac{1}{2})sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}} - \frac{tan(\frac{1}{2}x)*2sec^{2}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\\=&\frac{-2sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}(tan(\frac{1}{2}x) - 1)^{6}} + \frac{3sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{5}} - \frac{3tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{4}} - \frac{3sec^{6}(\frac{1}{2}x)}{2(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{4}} + \frac{3tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{3}} - \frac{sec^{4}(\frac{1}{2}x)}{2(tan(\frac{1}{2}x) - 1)^{2}(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{tan^{2}(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-2sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}(tan(\frac{1}{2}x) - 1)^{6}} + \frac{3sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{5}} - \frac{3tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{4}} - \frac{3sec^{6}(\frac{1}{2}x)}{2(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{4}} + \frac{3tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{3}} - \frac{sec^{4}(\frac{1}{2}x)}{2(tan(\frac{1}{2}x) - 1)^{2}(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{tan^{2}(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\right)}{dx}\\=&\frac{-2(\frac{-3(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{4}})sec^{6}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{6}} - \frac{2(\frac{-6(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{7}})sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}} - \frac{2*6sec^{6}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}(tan(\frac{1}{2}x) - 1)^{6}} + \frac{3(\frac{-2(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}})sec^{6}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{5}} + \frac{3(\frac{-5(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{6}})sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}} + \frac{3*6sec^{6}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{5}} - \frac{3(\frac{-2(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}})tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{4}} - \frac{3(\frac{-4(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{5}})tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}} - \frac{3sec^{2}(\frac{1}{2}x)(\frac{1}{2})sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{4}} - \frac{3tan(\frac{1}{2}x)*4sec^{4}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{4}} - \frac{3(\frac{-(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}})sec^{6}(\frac{1}{2}x)}{2(tan(\frac{1}{2}x) - 1)^{4}} - \frac{3(\frac{-4(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{5}})sec^{6}(\frac{1}{2}x)}{2(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{3*6sec^{6}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{2(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{4}} + \frac{3(\frac{-(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}})tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{3}} + \frac{3(\frac{-3(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{4}})tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} + \frac{3sec^{2}(\frac{1}{2}x)(\frac{1}{2})sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{3}} + \frac{3tan(\frac{1}{2}x)*4sec^{4}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{3}} - \frac{(\frac{-2(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{3}})sec^{4}(\frac{1}{2}x)}{2(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{(\frac{-(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}})sec^{4}(\frac{1}{2}x)}{2(tan(\frac{1}{2}x) - 1)^{2}} - \frac{4sec^{4}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{2(tan(\frac{1}{2}x) - 1)^{2}(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{(\frac{-(2(\frac{-(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{2}}) + 0)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}})tan^{2}(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{2}} - \frac{(\frac{-2(sec^{2}(\frac{1}{2}x)(\frac{1}{2}) + 0)}{(tan(\frac{1}{2}x) - 1)^{3}})tan^{2}(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{2tan(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)(\frac{1}{2})sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}} - \frac{tan^{2}(\frac{1}{2}x)*2sec^{2}(\frac{1}{2}x)tan(\frac{1}{2}x)*\frac{1}{2}}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\\=&\frac{-6sec^{8}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{4}(tan(\frac{1}{2}x) - 1)^{8}} + \frac{12sec^{8}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}(tan(\frac{1}{2}x) - 1)^{7}} - \frac{tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}} - \frac{9sec^{8}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{6}} + \frac{18tan(\frac{1}{2}x)sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{5}} - \frac{12tan(\frac{1}{2}x)sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{3}(tan(\frac{1}{2}x) - 1)^{6}} - \frac{2sec^{6}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{4}(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}} - \frac{7tan^{2}(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)^{2}(tan(\frac{1}{2}x) - 1)^{4}} + \frac{3sec^{8}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{5}} - \frac{9tan(\frac{1}{2}x)sec^{6}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{4}} + \frac{7tan^{2}(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{3}} + \frac{2sec^{6}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{3}(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{tan(\frac{1}{2}x)sec^{4}(\frac{1}{2}x)}{(tan(\frac{1}{2}x) - 1)^{2}(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)} - \frac{tan^{3}(\frac{1}{2}x)sec^{2}(\frac{1}{2}x)}{(\frac{2}{(tan(\frac{1}{2}x) - 1)} + 1)(tan(\frac{1}{2}x) - 1)^{2}}\\ \end{split}\end{equation} \]

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