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

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