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

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