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。
$$\begin{equation}\begin{split}【1/1】求函数\frac{e^{arcsin(x)}}{sqrt(1 - xx)} 关于 x 的 4 阶导数：\\\end{split}\end{equation}$$ $$\begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{e^{arcsin(x)}}{sqrt(-x^{2} + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{e^{arcsin(x)}}{sqrt(-x^{2} + 1)}\right)}{dx}\\=&\frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xe^{arcsin(x)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xe^{arcsin(x)}}{sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + (\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arcsin(x)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arcsin(x)} + \frac{3*2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})e^{arcsin(x)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}sqrt(-x^{2} + 1)} + \frac{9x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{3}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{10xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{2e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{15x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}sqrt(-x^{2} + 1)} + \frac{9x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{3}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{10xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{2e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{15x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arcsin(x)}}{sqrt(-x^{2} + 1)} + \frac{3*2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}sqrt(-x^{2} + 1)} + \frac{3x^{2}e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}sqrt(-x^{2} + 1)} + \frac{3x^{2}e^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})e^{arcsin(x)}}{sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{xe^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arcsin(x)}}{sqrt(-x^{2} + 1)} + \frac{2e^{arcsin(x)}}{(-x^{2} + 1)^{2}sqrt(-x^{2} + 1)} + \frac{2xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}sqrt(-x^{2} + 1)} + \frac{2xe^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{2}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + 9(\frac{-3(-2x + 0)}{(-x^{2} + 1)^{4}})x^{2}e^{arcsin(x)} + \frac{9*2xe^{arcsin(x)}}{(-x^{2} + 1)^{3}} + \frac{9x^{2}e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{3}} + \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}*-(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + 10(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})xe^{arcsin(x)} + \frac{10e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{10xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})e^{arcsin(x)} + \frac{2e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + 15(\frac{\frac{-7}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{9}{2}}})x^{3}e^{arcsin(x)} + \frac{15*3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{15x^{3}e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{3(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{3*2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}sqrt(-x^{2} + 1)} + \frac{12xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}sqrt(-x^{2} + 1)} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}sqrt(-x^{2} + 1)} + \frac{12x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{3}sqrt(-x^{2} + 1)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}sqrt(-x^{2} + 1)} + \frac{75x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{4}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{3e^{arcsin(x)}}{(-x^{2} + 1)^{2}sqrt(-x^{2} + 1)} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}sqrt(-x^{2} + 1)} + \frac{108x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{38xe^{arcsin(x)}}{(-x^{2} + 1)^{3}} + \frac{9x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{3}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}sqrt(-x^{2} + 1)} + \frac{12e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{10xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{2e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{105x^{4}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{9}{2}}} + \frac{15x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{7}{2}}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{3}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation}$$

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