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。
\[ \begin{equation}\begin{split}【1/1】求函数arcsin(sqrt(1 - 4{x}^{2})) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = arcsin(sqrt(-4x^{2} + 1))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arcsin(sqrt(-4x^{2} + 1))\right)}{dx}\\=&(\frac{(\frac{(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}})}{((1 - (sqrt(-4x^{2} + 1))^{2})^{\frac{1}{2}})})\\=&\frac{-4x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-4x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{-4(\frac{\frac{-1}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}})x}{(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{4(\frac{\frac{-1}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{3}{2}}})x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} - \frac{4}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{16x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{16x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{3}{2}}} - \frac{4}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{16x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{16x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{3}{2}}} - \frac{4}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{16(\frac{\frac{-3}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}})x^{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + \frac{16(\frac{\frac{-1}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{3}{2}}})x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} + \frac{16*2x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{16(\frac{\frac{-1}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}})x^{2}}{(-4x^{2} + 1)^{\frac{3}{2}}} - \frac{16(\frac{\frac{-3}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} - \frac{16*2x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{3}{2}}} - \frac{4(\frac{\frac{-1}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}})}{(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{4(\frac{\frac{-1}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{3}{2}}})}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-192x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} + \frac{128x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{3}{2}}} + \frac{32x}{(-4x^{2} + 1)^{\frac{1}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} - \frac{192x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{5}{2}}} - \frac{32x}{(-4x^{2} + 1)^{\frac{3}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} + \frac{16x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{16x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-192x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} + \frac{128x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{3}{2}}} + \frac{32x}{(-4x^{2} + 1)^{\frac{1}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} - \frac{192x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{5}{2}}} - \frac{32x}{(-4x^{2} + 1)^{\frac{3}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} + \frac{16x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{16x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{-192(\frac{\frac{-5}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{7}{2}}})x^{3}}{(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{192(\frac{\frac{-1}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{3}{2}}})x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}} - \frac{192*3x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} + \frac{128(\frac{\frac{-3}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}})x^{3}}{(-4x^{2} + 1)^{\frac{3}{2}}} + \frac{128(\frac{\frac{-3}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{5}{2}}})x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} + \frac{128*3x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{3}{2}}} + \frac{32(\frac{\frac{-1}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{3}{2}}})x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} + \frac{32(\frac{\frac{-3}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}})x}{(-4x^{2} + 1)^{\frac{1}{2}}} + \frac{32}{(-4x^{2} + 1)^{\frac{1}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} - \frac{192(\frac{\frac{-1}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}})x^{3}}{(-4x^{2} + 1)^{\frac{5}{2}}} - \frac{192(\frac{\frac{-5}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{7}{2}}})x^{3}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} - \frac{192*3x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{5}{2}}} - \frac{32(\frac{\frac{-3}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{5}{2}}})x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} - \frac{32(\frac{\frac{-1}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}})x}{(-4x^{2} + 1)^{\frac{3}{2}}} - \frac{32}{(-4x^{2} + 1)^{\frac{3}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} + \frac{16(\frac{\frac{-3}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}})x}{(-4x^{2} + 1)^{\frac{1}{2}}} + \frac{16(\frac{\frac{-1}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{3}{2}}})x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} + \frac{16}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{16(\frac{\frac{-1}{2}(\frac{-2(-4x^{2} + 1)^{\frac{1}{2}}(-4*2x + 0)*\frac{1}{2}}{(-4x^{2} + 1)^{\frac{1}{2}}} + 0)}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}})x}{(-4x^{2} + 1)^{\frac{3}{2}}} - \frac{16(\frac{\frac{-3}{2}(-4*2x + 0)}{(-4x^{2} + 1)^{\frac{5}{2}}})x}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} - \frac{16}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{3840x^{4}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{7}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{2304x^{4}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}(-4x^{2} + 1)^{\frac{3}{2}}} - \frac{576x^{2}}{(-4x^{2} + 1)^{\frac{1}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}} + \frac{2304x^{4}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{5}{2}}} + \frac{640x^{2}}{(-4x^{2} + 1)^{\frac{3}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} - \frac{576x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{5}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{3840x^{4}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{7}{2}}} - \frac{960x^{2}}{(-4x^{2} + 1)^{\frac{5}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} + \frac{128x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{3}{2}}} - \frac{192x^{2}}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{5}{2}}} + \frac{32}{(-4x^{2} + 1)^{\frac{1}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}} + \frac{16}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{3}{2}}(-4x^{2} + 1)^{\frac{1}{2}}} - \frac{32}{(-4x^{2} + 1)^{\frac{3}{2}}(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}} - \frac{16}{(-sqrt(-4x^{2} + 1)^{2} + 1)^{\frac{1}{2}}(-4x^{2} + 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]

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