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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{xtan(x)}{(ln(sin(x)))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{xtan(x)}{ln(sin(x))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{xtan(x)}{ln(sin(x))}\right)}{dx}\\=&\frac{tan(x)}{ln(sin(x))} + \frac{x*-cos(x)tan(x)}{ln^{2}(sin(x))(sin(x))} + \frac{xsec^{2}(x)(1)}{ln(sin(x))}\\=&\frac{tan(x)}{ln(sin(x))} - \frac{xcos(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{xsec^{2}(x)}{ln(sin(x))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{tan(x)}{ln(sin(x))} - \frac{xcos(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{xsec^{2}(x)}{ln(sin(x))}\right)}{dx}\\=&\frac{-cos(x)tan(x)}{ln^{2}(sin(x))(sin(x))} + \frac{sec^{2}(x)(1)}{ln(sin(x))} - \frac{cos(x)tan(x)}{ln^{2}(sin(x))sin(x)} - \frac{x*-2cos(x)cos(x)tan(x)}{ln^{3}(sin(x))(sin(x))sin(x)} - \frac{x*-cos(x)cos(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{x*-sin(x)tan(x)}{ln^{2}(sin(x))sin(x)} - \frac{xcos(x)sec^{2}(x)(1)}{ln^{2}(sin(x))sin(x)} + \frac{sec^{2}(x)}{ln(sin(x))} + \frac{x*-cos(x)sec^{2}(x)}{ln^{2}(sin(x))(sin(x))} + \frac{x*2sec^{2}(x)tan(x)}{ln(sin(x))}\\=&\frac{-2cos(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{2sec^{2}(x)}{ln(sin(x))} + \frac{2xcos^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{xcos^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{2xtan(x)sec^{2}(x)}{ln(sin(x))} - \frac{2xcos(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} + \frac{xtan(x)}{ln^{2}(sin(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2cos(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{2sec^{2}(x)}{ln(sin(x))} + \frac{2xcos^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{xcos^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{2xtan(x)sec^{2}(x)}{ln(sin(x))} - \frac{2xcos(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} + \frac{xtan(x)}{ln^{2}(sin(x))}\right)}{dx}\\=&\frac{-2*-2cos(x)cos(x)tan(x)}{ln^{3}(sin(x))(sin(x))sin(x)} - \frac{2*-cos(x)cos(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{2*-sin(x)tan(x)}{ln^{2}(sin(x))sin(x)} - \frac{2cos(x)sec^{2}(x)(1)}{ln^{2}(sin(x))sin(x)} + \frac{2*-cos(x)sec^{2}(x)}{ln^{2}(sin(x))(sin(x))} + \frac{2*2sec^{2}(x)tan(x)}{ln(sin(x))} + \frac{2cos^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{2x*-3cos(x)cos^{2}(x)tan(x)}{ln^{4}(sin(x))(sin(x))sin^{2}(x)} + \frac{2x*-2cos(x)cos^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{3}(x)} + \frac{2x*-2cos(x)sin(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{2xcos^{2}(x)sec^{2}(x)(1)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{cos^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{x*-2cos(x)cos^{2}(x)tan(x)}{ln^{3}(sin(x))(sin(x))sin^{2}(x)} + \frac{x*-2cos(x)cos^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{3}(x)} + \frac{x*-2cos(x)sin(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{xcos^{2}(x)sec^{2}(x)(1)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{2tan(x)sec^{2}(x)}{ln(sin(x))} + \frac{2x*-cos(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))(sin(x))} + \frac{2xsec^{2}(x)(1)sec^{2}(x)}{ln(sin(x))} + \frac{2xtan(x)*2sec^{2}(x)tan(x)}{ln(sin(x))} - \frac{2cos(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{2x*-2cos(x)cos(x)sec^{2}(x)}{ln^{3}(sin(x))(sin(x))sin(x)} - \frac{2x*-cos(x)cos(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{2x*-sin(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{2xcos(x)*2sec^{2}(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{tan(x)}{ln^{2}(sin(x))} + \frac{x*-2cos(x)tan(x)}{ln^{3}(sin(x))(sin(x))} + \frac{xsec^{2}(x)(1)}{ln^{2}(sin(x))}\\=&\frac{6cos^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{3cos^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{6tan(x)sec^{2}(x)}{ln(sin(x))} - \frac{6cos(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} + \frac{3tan(x)}{ln^{2}(sin(x))} - \frac{6xcos(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{6xcos^{3}(x)tan(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6xcos(x)tan(x)}{ln^{3}(sin(x))sin(x)} + \frac{6xcos^{2}(x)sec^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)} - \frac{2xcos^{3}(x)tan(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{2xcos(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{3xcos^{2}(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6xcos^{3}(x)tan(x)}{ln^{4}(sin(x))sin^{3}(x)} + \frac{2xsec^{4}(x)}{ln(sin(x))} + \frac{4xtan^{2}(x)sec^{2}(x)}{ln(sin(x))} + \frac{3xsec^{2}(x)}{ln^{2}(sin(x))}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{6cos^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{3cos^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{6tan(x)sec^{2}(x)}{ln(sin(x))} - \frac{6cos(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} + \frac{3tan(x)}{ln^{2}(sin(x))} - \frac{6xcos(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{6xcos^{3}(x)tan(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6xcos(x)tan(x)}{ln^{3}(sin(x))sin(x)} + \frac{6xcos^{2}(x)sec^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)} - \frac{2xcos^{3}(x)tan(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{2xcos(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{3xcos^{2}(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6xcos^{3}(x)tan(x)}{ln^{4}(sin(x))sin^{3}(x)} + \frac{2xsec^{4}(x)}{ln(sin(x))} + \frac{4xtan^{2}(x)sec^{2}(x)}{ln(sin(x))} + \frac{3xsec^{2}(x)}{ln^{2}(sin(x))}\right)}{dx}\\=&\frac{6*-3cos(x)cos^{2}(x)tan(x)}{ln^{4}(sin(x))(sin(x))sin^{2}(x)} + \frac{6*-2cos(x)cos^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{3}(x)} + \frac{6*-2cos(x)sin(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{6cos^{2}(x)sec^{2}(x)(1)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{3*-2cos(x)cos^{2}(x)tan(x)}{ln^{3}(sin(x))(sin(x))sin^{2}(x)} + \frac{3*-2cos(x)cos^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{3}(x)} + \frac{3*-2cos(x)sin(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{3cos^{2}(x)sec^{2}(x)(1)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{6*-cos(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))(sin(x))} + \frac{6sec^{2}(x)(1)sec^{2}(x)}{ln(sin(x))} + \frac{6tan(x)*2sec^{2}(x)tan(x)}{ln(sin(x))} - \frac{6*-2cos(x)cos(x)sec^{2}(x)}{ln^{3}(sin(x))(sin(x))sin(x)} - \frac{6*-cos(x)cos(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6*-sin(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{6cos(x)*2sec^{2}(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{3*-2cos(x)tan(x)}{ln^{3}(sin(x))(sin(x))} + \frac{3sec^{2}(x)(1)}{ln^{2}(sin(x))} - \frac{6cos(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{6x*-2cos(x)cos(x)tan(x)sec^{2}(x)}{ln^{3}(sin(x))(sin(x))sin(x)} - \frac{6x*-cos(x)cos(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6x*-sin(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{6xcos(x)sec^{2}(x)(1)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{6xcos(x)tan(x)*2sec^{2}(x)tan(x)}{ln^{2}(sin(x))sin(x)} - \frac{6cos^{3}(x)tan(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6x*-3cos(x)cos^{3}(x)tan(x)}{ln^{4}(sin(x))(sin(x))sin^{3}(x)} - \frac{6x*-3cos(x)cos^{3}(x)tan(x)}{ln^{3}(sin(x))sin^{4}(x)} - \frac{6x*-3cos^{2}(x)sin(x)tan(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6xcos^{3}(x)sec^{2}(x)(1)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6cos(x)tan(x)}{ln^{3}(sin(x))sin(x)} - \frac{6x*-3cos(x)cos(x)tan(x)}{ln^{4}(sin(x))(sin(x))sin(x)} - \frac{6x*-cos(x)cos(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} - \frac{6x*-sin(x)tan(x)}{ln^{3}(sin(x))sin(x)} - \frac{6xcos(x)sec^{2}(x)(1)}{ln^{3}(sin(x))sin(x)} + \frac{6cos^{2}(x)sec^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{6x*-3cos(x)cos^{2}(x)sec^{2}(x)}{ln^{4}(sin(x))(sin(x))sin^{2}(x)} + \frac{6x*-2cos(x)cos^{2}(x)sec^{2}(x)}{ln^{3}(sin(x))sin^{3}(x)} + \frac{6x*-2cos(x)sin(x)sec^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{6xcos^{2}(x)*2sec^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} - \frac{2cos^{3}(x)tan(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{2x*-2cos(x)cos^{3}(x)tan(x)}{ln^{3}(sin(x))(sin(x))sin^{3}(x)} - \frac{2x*-3cos(x)cos^{3}(x)tan(x)}{ln^{2}(sin(x))sin^{4}(x)} - \frac{2x*-3cos^{2}(x)sin(x)tan(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{2xcos^{3}(x)sec^{2}(x)(1)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{2cos(x)tan(x)}{ln^{2}(sin(x))sin(x)} - \frac{2x*-2cos(x)cos(x)tan(x)}{ln^{3}(sin(x))(sin(x))sin(x)} - \frac{2x*-cos(x)cos(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{2x*-sin(x)tan(x)}{ln^{2}(sin(x))sin(x)} - \frac{2xcos(x)sec^{2}(x)(1)}{ln^{2}(sin(x))sin(x)} + \frac{3cos^{2}(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{3x*-2cos(x)cos^{2}(x)sec^{2}(x)}{ln^{3}(sin(x))(sin(x))sin^{2}(x)} + \frac{3x*-2cos(x)cos^{2}(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{3}(x)} + \frac{3x*-2cos(x)sin(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{3xcos^{2}(x)*2sec^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6cos^{3}(x)tan(x)}{ln^{4}(sin(x))sin^{3}(x)} - \frac{6x*-4cos(x)cos^{3}(x)tan(x)}{ln^{5}(sin(x))(sin(x))sin^{3}(x)} - \frac{6x*-3cos(x)cos^{3}(x)tan(x)}{ln^{4}(sin(x))sin^{4}(x)} - \frac{6x*-3cos^{2}(x)sin(x)tan(x)}{ln^{4}(sin(x))sin^{3}(x)} - \frac{6xcos^{3}(x)sec^{2}(x)(1)}{ln^{4}(sin(x))sin^{3}(x)} + \frac{2sec^{4}(x)}{ln(sin(x))} + \frac{2x*-cos(x)sec^{4}(x)}{ln^{2}(sin(x))(sin(x))} + \frac{2x*4sec^{4}(x)tan(x)}{ln(sin(x))} + \frac{4tan^{2}(x)sec^{2}(x)}{ln(sin(x))} + \frac{4x*-cos(x)tan^{2}(x)sec^{2}(x)}{ln^{2}(sin(x))(sin(x))} + \frac{4x*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{ln(sin(x))} + \frac{4xtan^{2}(x)*2sec^{2}(x)tan(x)}{ln(sin(x))} + \frac{3sec^{2}(x)}{ln^{2}(sin(x))} + \frac{3x*-2cos(x)sec^{2}(x)}{ln^{3}(sin(x))(sin(x))} + \frac{3x*2sec^{2}(x)tan(x)}{ln^{2}(sin(x))}\\=&\frac{-24cos(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} - \frac{24cos^{3}(x)tan(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{24cos(x)tan(x)}{ln^{3}(sin(x))sin(x)} + \frac{24cos^{2}(x)sec^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)} - \frac{8cos^{3}(x)tan(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{8cos(x)tan(x)}{ln^{2}(sin(x))sin(x)} + \frac{12cos^{2}(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{24cos^{3}(x)tan(x)}{ln^{4}(sin(x))sin^{3}(x)} + \frac{8sec^{4}(x)}{ln(sin(x))} + \frac{16tan^{2}(x)sec^{2}(x)}{ln(sin(x))} + \frac{12sec^{2}(x)}{ln^{2}(sin(x))} + \frac{24xcos^{2}(x)tan(x)sec^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)} + \frac{12xcos^{2}(x)tan(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{12xtan(x)sec^{2}(x)}{ln^{2}(sin(x))} - \frac{8xcos(x)sec^{4}(x)}{ln^{2}(sin(x))sin(x)} - \frac{16xcos(x)tan^{2}(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} + \frac{36xcos^{4}(x)tan(x)}{ln^{4}(sin(x))sin^{4}(x)} + \frac{22xcos^{4}(x)tan(x)}{ln^{3}(sin(x))sin^{4}(x)} + \frac{28xcos^{2}(x)tan(x)}{ln^{3}(sin(x))sin^{2}(x)} - \frac{24xcos^{3}(x)sec^{2}(x)}{ln^{3}(sin(x))sin^{3}(x)} + \frac{36xcos^{2}(x)tan(x)}{ln^{4}(sin(x))sin^{2}(x)} + \frac{16xtan(x)sec^{4}(x)}{ln(sin(x))} - \frac{24xcos(x)sec^{2}(x)}{ln^{3}(sin(x))sin(x)} - \frac{24xcos^{3}(x)sec^{2}(x)}{ln^{4}(sin(x))sin^{3}(x)} + \frac{6xcos^{4}(x)tan(x)}{ln^{2}(sin(x))sin^{4}(x)} + \frac{8xcos^{2}(x)tan(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{8xcos^{3}(x)sec^{2}(x)}{ln^{2}(sin(x))sin^{3}(x)} + \frac{8xtan^{3}(x)sec^{2}(x)}{ln(sin(x))} - \frac{8xcos(x)sec^{2}(x)}{ln^{2}(sin(x))sin(x)} + \frac{24xcos^{4}(x)tan(x)}{ln^{5}(sin(x))sin^{4}(x)} + \frac{6xtan(x)}{ln^{3}(sin(x))} + \frac{2xtan(x)}{ln^{2}(sin(x))}\\ \end{split}\end{equation} \]

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