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。
\[ \begin{equation}\begin{split}【1/1】求函数arctan(sqrt(xx - 1)) - \frac{log_{3}^{x}}{sqrt(xx - 1)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = arctan(sqrt(x^{2} - 1)) - \frac{log_{3}^{x}}{sqrt(x^{2} - 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arctan(sqrt(x^{2} - 1)) - \frac{log_{3}^{x}}{sqrt(x^{2} - 1)}\right)}{dx}\\=&(\frac{(\frac{(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}})}{(1 + (sqrt(x^{2} - 1))^{2})}) - \frac{(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{3}^{x}}{(3)})}{(ln(3))})}{sqrt(x^{2} - 1)} - \frac{log_{3}^{x}*-(2x + 0)*\frac{1}{2}}{(x^{2} - 1)(x^{2} - 1)^{\frac{1}{2}}}\\=&\frac{xlog_{3}^{x}}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{1}{xln(3)sqrt(x^{2} - 1)} + \frac{x}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{xlog_{3}^{x}}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{1}{xln(3)sqrt(x^{2} - 1)} + \frac{x}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)}\right)}{dx}\\=&(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})xlog_{3}^{x} + \frac{log_{3}^{x}}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{x(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{3}^{x}}{(3)})}{(ln(3))})}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{-1}{x^{2}ln(3)sqrt(x^{2} - 1)} - \frac{-0}{xln^{2}(3)(3)sqrt(x^{2} - 1)} - \frac{-(2x + 0)*\frac{1}{2}}{xln(3)(x^{2} - 1)(x^{2} - 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x}{(sqrt(x^{2} - 1)^{2} + 1)} + \frac{(\frac{-(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{2}})x}{(x^{2} - 1)^{\frac{1}{2}}} + \frac{1}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)}\\=& - \frac{3x^{2}log_{3}^{x}}{(x^{2} - 1)^{\frac{5}{2}}} + \frac{log_{3}^{x}}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{2}{(x^{2} - 1)^{\frac{3}{2}}ln(3)} + \frac{1}{x^{2}ln(3)sqrt(x^{2} - 1)} - \frac{x^{2}}{(x^{2} - 1)^{\frac{3}{2}}(sqrt(x^{2} - 1)^{2} + 1)} - \frac{2x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{1}{2}}} + \frac{1}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{3x^{2}log_{3}^{x}}{(x^{2} - 1)^{\frac{5}{2}}} + \frac{log_{3}^{x}}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{2}{(x^{2} - 1)^{\frac{3}{2}}ln(3)} + \frac{1}{x^{2}ln(3)sqrt(x^{2} - 1)} - \frac{x^{2}}{(x^{2} - 1)^{\frac{3}{2}}(sqrt(x^{2} - 1)^{2} + 1)} - \frac{2x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{1}{2}}} + \frac{1}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)}\right)}{dx}\\=& - 3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} - 1)^{\frac{7}{2}}})x^{2}log_{3}^{x} - \frac{3*2xlog_{3}^{x}}{(x^{2} - 1)^{\frac{5}{2}}} - \frac{3x^{2}(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{3}^{x}}{(3)})}{(ln(3))})}{(x^{2} - 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})log_{3}^{x} + \frac{(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{3}^{x}}{(3)})}{(ln(3))})}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})}{ln(3)} + \frac{2*-0}{(x^{2} - 1)^{\frac{3}{2}}ln^{2}(3)(3)} + \frac{-2}{x^{3}ln(3)sqrt(x^{2} - 1)} + \frac{-0}{x^{2}ln^{2}(3)(3)sqrt(x^{2} - 1)} + \frac{-(2x + 0)*\frac{1}{2}}{x^{2}ln(3)(x^{2} - 1)(x^{2} - 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)} - \frac{(\frac{-(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{2}})x^{2}}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{2x}{(x^{2} - 1)^{\frac{3}{2}}(sqrt(x^{2} - 1)^{2} + 1)} - \frac{2(\frac{-2(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{3}})x^{2}}{(x^{2} - 1)^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}} - \frac{2*2x}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})}{(sqrt(x^{2} - 1)^{2} + 1)} + \frac{(\frac{-(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{2}})}{(x^{2} - 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}log_{3}^{x}}{(x^{2} - 1)^{\frac{7}{2}}} - \frac{9xlog_{3}^{x}}{(x^{2} - 1)^{\frac{5}{2}}} - \frac{9x}{(x^{2} - 1)^{\frac{5}{2}}ln(3)} - \frac{2}{x^{3}ln(3)sqrt(x^{2} - 1)} + \frac{3x^{3}}{(x^{2} - 1)^{\frac{5}{2}}(sqrt(x^{2} - 1)^{2} + 1)} + \frac{4x^{3}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{3}{2}}} - \frac{2x}{(sqrt(x^{2} - 1)^{2} + 1)(x^{2} - 1)^{\frac{3}{2}}} + \frac{8x^{3}}{(sqrt(x^{2} - 1)^{2} + 1)^{3}(x^{2} - 1)^{\frac{1}{2}}} - \frac{4x}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)^{2}} - \frac{x}{(x^{2} - 1)^{\frac{3}{2}}(sqrt(x^{2} - 1)^{2} + 1)} - \frac{2x}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{15x^{3}log_{3}^{x}}{(x^{2} - 1)^{\frac{7}{2}}} - \frac{9xlog_{3}^{x}}{(x^{2} - 1)^{\frac{5}{2}}} - \frac{9x}{(x^{2} - 1)^{\frac{5}{2}}ln(3)} - \frac{2}{x^{3}ln(3)sqrt(x^{2} - 1)} + \frac{3x^{3}}{(x^{2} - 1)^{\frac{5}{2}}(sqrt(x^{2} - 1)^{2} + 1)} + \frac{4x^{3}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{3}{2}}} - \frac{2x}{(sqrt(x^{2} - 1)^{2} + 1)(x^{2} - 1)^{\frac{3}{2}}} + \frac{8x^{3}}{(sqrt(x^{2} - 1)^{2} + 1)^{3}(x^{2} - 1)^{\frac{1}{2}}} - \frac{4x}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)^{2}} - \frac{x}{(x^{2} - 1)^{\frac{3}{2}}(sqrt(x^{2} - 1)^{2} + 1)} - \frac{2x}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{1}{2}}}\right)}{dx}\\=&15(\frac{\frac{-7}{2}(2x + 0)}{(x^{2} - 1)^{\frac{9}{2}}})x^{3}log_{3}^{x} + \frac{15*3x^{2}log_{3}^{x}}{(x^{2} - 1)^{\frac{7}{2}}} + \frac{15x^{3}(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{3}^{x}}{(3)})}{(ln(3))})}{(x^{2} - 1)^{\frac{7}{2}}} - 9(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} - 1)^{\frac{7}{2}}})xlog_{3}^{x} - \frac{9log_{3}^{x}}{(x^{2} - 1)^{\frac{5}{2}}} - \frac{9x(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{3}^{x}}{(3)})}{(ln(3))})}{(x^{2} - 1)^{\frac{5}{2}}} - \frac{9(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} - 1)^{\frac{7}{2}}})x}{ln(3)} - \frac{9}{(x^{2} - 1)^{\frac{5}{2}}ln(3)} - \frac{9x*-0}{(x^{2} - 1)^{\frac{5}{2}}ln^{2}(3)(3)} - \frac{2*-3}{x^{4}ln(3)sqrt(x^{2} - 1)} - \frac{2*-0}{x^{3}ln^{2}(3)(3)sqrt(x^{2} - 1)} - \frac{2*-(2x + 0)*\frac{1}{2}}{x^{3}ln(3)(x^{2} - 1)(x^{2} - 1)^{\frac{1}{2}}} + \frac{3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} - 1)^{\frac{7}{2}}})x^{3}}{(sqrt(x^{2} - 1)^{2} + 1)} + \frac{3(\frac{-(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{2}})x^{3}}{(x^{2} - 1)^{\frac{5}{2}}} + \frac{3*3x^{2}}{(x^{2} - 1)^{\frac{5}{2}}(sqrt(x^{2} - 1)^{2} + 1)} + \frac{4(\frac{-2(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{3}})x^{3}}{(x^{2} - 1)^{\frac{3}{2}}} + \frac{4(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x^{3}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}} + \frac{4*3x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{3}{2}}} - \frac{2(\frac{-(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{2}})x}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x}{(sqrt(x^{2} - 1)^{2} + 1)} - \frac{2}{(sqrt(x^{2} - 1)^{2} + 1)(x^{2} - 1)^{\frac{3}{2}}} + \frac{8(\frac{-3(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{4}})x^{3}}{(x^{2} - 1)^{\frac{1}{2}}} + \frac{8(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x^{3}}{(sqrt(x^{2} - 1)^{2} + 1)^{3}} + \frac{8*3x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)^{3}(x^{2} - 1)^{\frac{1}{2}}} - \frac{4(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} - 1)^{\frac{3}{2}}})x}{(sqrt(x^{2} - 1)^{2} + 1)^{2}} - \frac{4(\frac{-2(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{3}})x}{(x^{2} - 1)^{\frac{1}{2}}} - \frac{4}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)^{2}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} - 1)^{\frac{5}{2}}})x}{(sqrt(x^{2} - 1)^{2} + 1)} - \frac{(\frac{-(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{2} + 1)^{2}})x}{(x^{2} - 1)^{\frac{3}{2}}} - \frac{1}{(x^{2} - 1)^{\frac{3}{2}}(sqrt(x^{2} - 1)^{2} + 1)} - \frac{2(\frac{-2(\frac{2(x^{2} - 1)^{\frac{1}{2}}(2x + 0)*\frac{1}{2}}{(x^{2} - 1)^{\frac{1}{2}}} + 0)}{(sqrt(x^{2} - 1)^{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}{(sqrt(x^{2} - 1)^{2} + 1)^{2}} - \frac{2}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{1}{2}}}\\=& - \frac{105x^{4}log_{3}^{x}}{(x^{2} - 1)^{\frac{9}{2}}} + \frac{90x^{2}log_{3}^{x}}{(x^{2} - 1)^{\frac{7}{2}}} + \frac{60x^{2}}{(x^{2} - 1)^{\frac{7}{2}}ln(3)} - \frac{9log_{3}^{x}}{(x^{2} - 1)^{\frac{5}{2}}} - \frac{18}{(x^{2} - 1)^{\frac{5}{2}}ln(3)} + \frac{6}{x^{4}ln(3)sqrt(x^{2} - 1)} + \frac{2}{(x^{2} - 1)^{\frac{3}{2}}x^{2}ln(3)} - \frac{15x^{4}}{(x^{2} - 1)^{\frac{7}{2}}(sqrt(x^{2} - 1)^{2} + 1)} - \frac{18x^{4}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{5}{2}}} + \frac{9x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)(x^{2} - 1)^{\frac{5}{2}}} - \frac{24x^{4}}{(sqrt(x^{2} - 1)^{2} + 1)^{3}(x^{2} - 1)^{\frac{3}{2}}} + \frac{16x^{2}}{(x^{2} - 1)^{\frac{3}{2}}(sqrt(x^{2} - 1)^{2} + 1)^{2}} + \frac{8x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{3}{2}}} + \frac{9x^{2}}{(x^{2} - 1)^{\frac{5}{2}}(sqrt(x^{2} - 1)^{2} + 1)} + \frac{24x^{2}}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)^{3}} - \frac{48x^{4}}{(sqrt(x^{2} - 1)^{2} + 1)^{4}(x^{2} - 1)^{\frac{1}{2}}} + \frac{24x^{2}}{(sqrt(x^{2} - 1)^{2} + 1)^{3}(x^{2} - 1)^{\frac{1}{2}}} - \frac{4}{(x^{2} - 1)^{\frac{1}{2}}(sqrt(x^{2} - 1)^{2} + 1)^{2}} - \frac{2}{(sqrt(x^{2} - 1)^{2} + 1)^{2}(x^{2} - 1)^{\frac{1}{2}}} - \frac{1}{(x^{2} - 1)^{\frac{3}{2}}(sqrt(x^{2} - 1)^{2} + 1)} - \frac{2}{(sqrt(x^{2} - 1)^{2} + 1)(x^{2} - 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]

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