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。
\[ \begin{equation}\begin{split}【1/1】求函数e^{2x}arcsin(3x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{2x}arcsin(3x)\right)}{dx}\\=&e^{2x}*2arcsin(3x) + e^{2x}(\frac{(3)}{((1 - (3x)^{2})^{\frac{1}{2}})})\\=&2e^{2x}arcsin(3x) + \frac{3e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( 2e^{2x}arcsin(3x) + \frac{3e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&2e^{2x}*2arcsin(3x) + 2e^{2x}(\frac{(3)}{((1 - (3x)^{2})^{\frac{1}{2}})}) + 3(\frac{\frac{-1}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{3}{2}}})e^{2x} + \frac{3e^{2x}*2}{(-9x^{2} + 1)^{\frac{1}{2}}}\\=&4e^{2x}arcsin(3x) + \frac{6e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}} + \frac{27xe^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}} + \frac{6e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 4e^{2x}arcsin(3x) + \frac{6e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}} + \frac{27xe^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}} + \frac{6e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&4e^{2x}*2arcsin(3x) + 4e^{2x}(\frac{(3)}{((1 - (3x)^{2})^{\frac{1}{2}})}) + 6(\frac{\frac{-1}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{3}{2}}})e^{2x} + \frac{6e^{2x}*2}{(-9x^{2} + 1)^{\frac{1}{2}}} + 27(\frac{\frac{-3}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{5}{2}}})xe^{2x} + \frac{27e^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}} + \frac{27xe^{2x}*2}{(-9x^{2} + 1)^{\frac{3}{2}}} + 6(\frac{\frac{-1}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{3}{2}}})e^{2x} + \frac{6e^{2x}*2}{(-9x^{2} + 1)^{\frac{1}{2}}}\\=&8e^{2x}arcsin(3x) + \frac{12e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}} + \frac{162xe^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}} + \frac{24e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}} + \frac{729x^{2}e^{2x}}{(-9x^{2} + 1)^{\frac{5}{2}}} + \frac{27e^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 8e^{2x}arcsin(3x) + \frac{12e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}} + \frac{162xe^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}} + \frac{24e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}} + \frac{729x^{2}e^{2x}}{(-9x^{2} + 1)^{\frac{5}{2}}} + \frac{27e^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&8e^{2x}*2arcsin(3x) + 8e^{2x}(\frac{(3)}{((1 - (3x)^{2})^{\frac{1}{2}})}) + 12(\frac{\frac{-1}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{3}{2}}})e^{2x} + \frac{12e^{2x}*2}{(-9x^{2} + 1)^{\frac{1}{2}}} + 162(\frac{\frac{-3}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{5}{2}}})xe^{2x} + \frac{162e^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}} + \frac{162xe^{2x}*2}{(-9x^{2} + 1)^{\frac{3}{2}}} + 24(\frac{\frac{-1}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{3}{2}}})e^{2x} + \frac{24e^{2x}*2}{(-9x^{2} + 1)^{\frac{1}{2}}} + 729(\frac{\frac{-5}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{2x} + \frac{729*2xe^{2x}}{(-9x^{2} + 1)^{\frac{5}{2}}} + \frac{729x^{2}e^{2x}*2}{(-9x^{2} + 1)^{\frac{5}{2}}} + 27(\frac{\frac{-3}{2}(-9*2x + 0)}{(-9x^{2} + 1)^{\frac{5}{2}}})e^{2x} + \frac{27e^{2x}*2}{(-9x^{2} + 1)^{\frac{3}{2}}}\\=&16e^{2x}arcsin(3x) + \frac{24e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}} + \frac{648xe^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}} + \frac{72e^{2x}}{(-9x^{2} + 1)^{\frac{1}{2}}} + \frac{5832x^{2}e^{2x}}{(-9x^{2} + 1)^{\frac{5}{2}}} + \frac{216e^{2x}}{(-9x^{2} + 1)^{\frac{3}{2}}} + \frac{32805x^{3}e^{2x}}{(-9x^{2} + 1)^{\frac{7}{2}}} + \frac{2187xe^{2x}}{(-9x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]

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