本次共计算 1 个题目：每一题对 x 求 3 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(tan(x) - x)}{(x - sin(x))} 关于 x 的 3 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{tan(x)}{(x - sin(x))} - \frac{x}{(x - sin(x))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{tan(x)}{(x - sin(x))} - \frac{x}{(x - sin(x))}\right)}{dx}\\=&(\frac{-(1 - cos(x))}{(x - sin(x))^{2}})tan(x) + \frac{sec^{2}(x)(1)}{(x - sin(x))} - (\frac{-(1 - cos(x))}{(x - sin(x))^{2}})x - \frac{1}{(x - sin(x))}\\=&\frac{cos(x)tan(x)}{(x - sin(x))^{2}} - \frac{tan(x)}{(x - sin(x))^{2}} + \frac{sec^{2}(x)}{(x - sin(x))} - \frac{xcos(x)}{(x - sin(x))^{2}} + \frac{x}{(x - sin(x))^{2}} - \frac{1}{(x - sin(x))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{cos(x)tan(x)}{(x - sin(x))^{2}} - \frac{tan(x)}{(x - sin(x))^{2}} + \frac{sec^{2}(x)}{(x - sin(x))} - \frac{xcos(x)}{(x - sin(x))^{2}} + \frac{x}{(x - sin(x))^{2}} - \frac{1}{(x - sin(x))}\right)}{dx}\\=&(\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})cos(x)tan(x) + \frac{-sin(x)tan(x)}{(x - sin(x))^{2}} + \frac{cos(x)sec^{2}(x)(1)}{(x - sin(x))^{2}} - (\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})tan(x) - \frac{sec^{2}(x)(1)}{(x - sin(x))^{2}} + (\frac{-(1 - cos(x))}{(x - sin(x))^{2}})sec^{2}(x) + \frac{2sec^{2}(x)tan(x)}{(x - sin(x))} - (\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})xcos(x) - \frac{cos(x)}{(x - sin(x))^{2}} - \frac{x*-sin(x)}{(x - sin(x))^{2}} + (\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})x + \frac{1}{(x - sin(x))^{2}} - (\frac{-(1 - cos(x))}{(x - sin(x))^{2}})\\=&\frac{2cos^{2}(x)tan(x)}{(x - sin(x))^{3}} - \frac{4cos(x)tan(x)}{(x - sin(x))^{3}} - \frac{sin(x)tan(x)}{(x - sin(x))^{2}} + \frac{2cos(x)sec^{2}(x)}{(x - sin(x))^{2}} + \frac{2tan(x)sec^{2}(x)}{(x - sin(x))} - \frac{2sec^{2}(x)}{(x - sin(x))^{2}} + \frac{2tan(x)}{(x - sin(x))^{3}} - \frac{2xcos^{2}(x)}{(x - sin(x))^{3}} + \frac{4xcos(x)}{(x - sin(x))^{3}} - \frac{2cos(x)}{(x - sin(x))^{2}} + \frac{xsin(x)}{(x - sin(x))^{2}} - \frac{2x}{(x - sin(x))^{3}} + \frac{2}{(x - sin(x))^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2cos^{2}(x)tan(x)}{(x - sin(x))^{3}} - \frac{4cos(x)tan(x)}{(x - sin(x))^{3}} - \frac{sin(x)tan(x)}{(x - sin(x))^{2}} + \frac{2cos(x)sec^{2}(x)}{(x - sin(x))^{2}} + \frac{2tan(x)sec^{2}(x)}{(x - sin(x))} - \frac{2sec^{2}(x)}{(x - sin(x))^{2}} + \frac{2tan(x)}{(x - sin(x))^{3}} - \frac{2xcos^{2}(x)}{(x - sin(x))^{3}} + \frac{4xcos(x)}{(x - sin(x))^{3}} - \frac{2cos(x)}{(x - sin(x))^{2}} + \frac{xsin(x)}{(x - sin(x))^{2}} - \frac{2x}{(x - sin(x))^{3}} + \frac{2}{(x - sin(x))^{2}}\right)}{dx}\\=&2(\frac{-3(1 - cos(x))}{(x - sin(x))^{4}})cos^{2}(x)tan(x) + \frac{2*-2cos(x)sin(x)tan(x)}{(x - sin(x))^{3}} + \frac{2cos^{2}(x)sec^{2}(x)(1)}{(x - sin(x))^{3}} - 4(\frac{-3(1 - cos(x))}{(x - sin(x))^{4}})cos(x)tan(x) - \frac{4*-sin(x)tan(x)}{(x - sin(x))^{3}} - \frac{4cos(x)sec^{2}(x)(1)}{(x - sin(x))^{3}} - (\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})sin(x)tan(x) - \frac{cos(x)tan(x)}{(x - sin(x))^{2}} - \frac{sin(x)sec^{2}(x)(1)}{(x - sin(x))^{2}} + 2(\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})cos(x)sec^{2}(x) + \frac{2*-sin(x)sec^{2}(x)}{(x - sin(x))^{2}} + \frac{2cos(x)*2sec^{2}(x)tan(x)}{(x - sin(x))^{2}} + 2(\frac{-(1 - cos(x))}{(x - sin(x))^{2}})tan(x)sec^{2}(x) + \frac{2sec^{2}(x)(1)sec^{2}(x)}{(x - sin(x))} + \frac{2tan(x)*2sec^{2}(x)tan(x)}{(x - sin(x))} - 2(\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})sec^{2}(x) - \frac{2*2sec^{2}(x)tan(x)}{(x - sin(x))^{2}} + 2(\frac{-3(1 - cos(x))}{(x - sin(x))^{4}})tan(x) + \frac{2sec^{2}(x)(1)}{(x - sin(x))^{3}} - 2(\frac{-3(1 - cos(x))}{(x - sin(x))^{4}})xcos^{2}(x) - \frac{2cos^{2}(x)}{(x - sin(x))^{3}} - \frac{2x*-2cos(x)sin(x)}{(x - sin(x))^{3}} + 4(\frac{-3(1 - cos(x))}{(x - sin(x))^{4}})xcos(x) + \frac{4cos(x)}{(x - sin(x))^{3}} + \frac{4x*-sin(x)}{(x - sin(x))^{3}} - 2(\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})cos(x) - \frac{2*-sin(x)}{(x - sin(x))^{2}} + (\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})xsin(x) + \frac{sin(x)}{(x - sin(x))^{2}} + \frac{xcos(x)}{(x - sin(x))^{2}} - 2(\frac{-3(1 - cos(x))}{(x - sin(x))^{4}})x - \frac{2}{(x - sin(x))^{3}} + 2(\frac{-2(1 - cos(x))}{(x - sin(x))^{3}})\\=&\frac{6cos(x)tan(x)sec^{2}(x)}{(x - sin(x))^{2}} - \frac{18cos^{2}(x)tan(x)}{(x - sin(x))^{4}} - \frac{6sin(x)cos(x)tan(x)}{(x - sin(x))^{3}} + \frac{6cos^{2}(x)sec^{2}(x)}{(x - sin(x))^{3}} + \frac{18cos(x)tan(x)}{(x - sin(x))^{4}} + \frac{6sin(x)tan(x)}{(x - sin(x))^{3}} - \frac{12cos(x)sec^{2}(x)}{(x - sin(x))^{3}} + \frac{6cos^{3}(x)tan(x)}{(x - sin(x))^{4}} - \frac{3sin(x)sec^{2}(x)}{(x - sin(x))^{2}} - \frac{cos(x)tan(x)}{(x - sin(x))^{2}} - \frac{6tan(x)sec^{2}(x)}{(x - sin(x))^{2}} + \frac{2sec^{4}(x)}{(x - sin(x))} + \frac{4tan^{2}(x)sec^{2}(x)}{(x - sin(x))} + \frac{6sec^{2}(x)}{(x - sin(x))^{3}} - \frac{6tan(x)}{(x - sin(x))^{4}} - \frac{6xcos^{3}(x)}{(x - sin(x))^{4}} + \frac{18xcos^{2}(x)}{(x - sin(x))^{4}} - \frac{6cos^{2}(x)}{(x - sin(x))^{3}} + \frac{6xsin(x)cos(x)}{(x - sin(x))^{3}} - \frac{18xcos(x)}{(x - sin(x))^{4}} + \frac{12cos(x)}{(x - sin(x))^{3}} - \frac{6xsin(x)}{(x - sin(x))^{3}} + \frac{3sin(x)}{(x - sin(x))^{2}} + \frac{xcos(x)}{(x - sin(x))^{2}} + \frac{6x}{(x - sin(x))^{4}} - \frac{6}{(x - sin(x))^{3}}\\ \end{split}\end{equation} \]

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