本次共计算 1 个题目：每一题对 x 求 2 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数8sqrt(-{x}^{2} + 1){\frac{1}{(2x + 1)}}^{3} - \frac{{x}^{2}}{((2x + 1){(-{x}^{2} + 1)}^{(\frac{3}{2})})} + \frac{4x}{(({(2x + 1)}^{2})sqrt(-{x}^{2} + 1))} - \frac{1}{((2x + 1)sqrt(-{x}^{2} + 1))} 关于 x 的 2 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{8sqrt(-x^{2} + 1)}{(2x + 1)^{3}} + \frac{4x}{(2x + 1)^{2}sqrt(-x^{2} + 1)} - \frac{x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})} - \frac{1}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{8sqrt(-x^{2} + 1)}{(2x + 1)^{3}} + \frac{4x}{(2x + 1)^{2}sqrt(-x^{2} + 1)} - \frac{x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})} - \frac{1}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))}\right)}{dx}\\=&8(\frac{-3(2 + 0)}{(2x + 1)^{4}})sqrt(-x^{2} + 1) + \frac{8(-2x + 0)*\frac{1}{2}}{(2x + 1)^{3}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{4(\frac{-2(2 + 0)}{(2x + 1)^{3}})x}{sqrt(-x^{2} + 1)} + \frac{4}{(2x + 1)^{2}sqrt(-x^{2} + 1)} + \frac{4x*-(-2x + 0)*\frac{1}{2}}{(2x + 1)^{2}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{-(2(\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0))x + 2(-x^{2} + 1)^{\frac{3}{2}} + (\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0)))}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}})x^{2} - \frac{2x}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})} - (\frac{-(2sqrt(-x^{2} + 1) + \frac{2x(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}})}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}})\\=&\frac{-48sqrt(-x^{2} + 1)}{(2x + 1)^{4}} - \frac{8x}{(2x + 1)^{3}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{16x}{(2x + 1)^{3}sqrt(-x^{2} + 1)} + \frac{4}{(2x + 1)^{2}sqrt(-x^{2} + 1)} + \frac{4x^{2}}{(-x^{2} + 1)^{\frac{3}{2}}(2x + 1)^{2}} - \frac{6(-x^{2} + 1)^{\frac{1}{2}}x^{4}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{3(-x^{2} + 1)^{\frac{1}{2}}x^{3}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} + \frac{2(-x^{2} + 1)^{\frac{3}{2}}x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{2x}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})} + \frac{2sqrt(-x^{2} + 1)}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}} - \frac{2x^{2}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{x}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-48sqrt(-x^{2} + 1)}{(2x + 1)^{4}} - \frac{8x}{(2x + 1)^{3}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{16x}{(2x + 1)^{3}sqrt(-x^{2} + 1)} + \frac{4}{(2x + 1)^{2}sqrt(-x^{2} + 1)} + \frac{4x^{2}}{(-x^{2} + 1)^{\frac{3}{2}}(2x + 1)^{2}} - \frac{6(-x^{2} + 1)^{\frac{1}{2}}x^{4}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{3(-x^{2} + 1)^{\frac{1}{2}}x^{3}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} + \frac{2(-x^{2} + 1)^{\frac{3}{2}}x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{2x}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})} + \frac{2sqrt(-x^{2} + 1)}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}} - \frac{2x^{2}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{x}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-48(\frac{-4(2 + 0)}{(2x + 1)^{5}})sqrt(-x^{2} + 1) - \frac{48(-2x + 0)*\frac{1}{2}}{(2x + 1)^{4}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{8(\frac{-3(2 + 0)}{(2x + 1)^{4}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{8(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(2x + 1)^{3}} - \frac{8}{(2x + 1)^{3}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{16(\frac{-3(2 + 0)}{(2x + 1)^{4}})x}{sqrt(-x^{2} + 1)} - \frac{16}{(2x + 1)^{3}sqrt(-x^{2} + 1)} - \frac{16x*-(-2x + 0)*\frac{1}{2}}{(2x + 1)^{3}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + \frac{4(\frac{-2(2 + 0)}{(2x + 1)^{3}})}{sqrt(-x^{2} + 1)} + \frac{4*-(-2x + 0)*\frac{1}{2}}{(2x + 1)^{2}(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}} + \frac{4(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(2x + 1)^{2}} + \frac{4(\frac{-2(2 + 0)}{(2x + 1)^{3}})x^{2}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{4*2x}{(-x^{2} + 1)^{\frac{3}{2}}(2x + 1)^{2}} - 6(\frac{-2(2(\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0))x + 2(-x^{2} + 1)^{\frac{3}{2}} + (\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0)))}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}})(-x^{2} + 1)^{\frac{1}{2}}x^{4} - \frac{6(\frac{\frac{1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{1}{2}}})x^{4}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{6(-x^{2} + 1)^{\frac{1}{2}}*4x^{3}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - 3(\frac{-2(2(\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0))x + 2(-x^{2} + 1)^{\frac{3}{2}} + (\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0)))}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}})(-x^{2} + 1)^{\frac{1}{2}}x^{3} - \frac{3(\frac{\frac{1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{1}{2}}})x^{3}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{3(-x^{2} + 1)^{\frac{1}{2}}*3x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} + 2(\frac{-2(2(\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0))x + 2(-x^{2} + 1)^{\frac{3}{2}} + (\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0)))}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}})(-x^{2} + 1)^{\frac{3}{2}}x^{2} + \frac{2(\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0))x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} + \frac{2(-x^{2} + 1)^{\frac{3}{2}}*2x}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - 2(\frac{-(2(\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0))x + 2(-x^{2} + 1)^{\frac{3}{2}} + (\frac{3}{2}(-x^{2} + 1)^{\frac{1}{2}}(-2x + 0)))}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}})x - \frac{2}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})} + 2(\frac{-2(2sqrt(-x^{2} + 1) + \frac{2x(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}})}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{3}})sqrt(-x^{2} + 1) + \frac{2(-2x + 0)*\frac{1}{2}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{2(\frac{-2(2sqrt(-x^{2} + 1) + \frac{2x(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}})}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{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}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}} - \frac{2*2x}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{-2(2sqrt(-x^{2} + 1) + \frac{2x(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(-2x + 0)*\frac{1}{2}}{(-x^{2} + 1)^{\frac{1}{2}}})}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{3}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}} - \frac{1}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{384sqrt(-x^{2} + 1)}{(2x + 1)^{5}} + \frac{96x}{(2x + 1)^{4}sqrt(-x^{2} + 1)} + \frac{16x^{2}sqrt(-x^{2} + 1)}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{3}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{8xsqrt(-x^{2} + 1)}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{3}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{24x^{2}}{(2x + 1)^{3}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{16x^{2}}{(-x^{2} + 1)^{\frac{3}{2}}(2x + 1)^{3}} - \frac{32}{(2x + 1)^{3}sqrt(-x^{2} + 1)} + \frac{4x}{(-x^{2} + 1)^{\frac{3}{2}}(2x + 1)^{2}} + \frac{6x^{5}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{8x}{(2x + 1)^{2}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{4}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{72x^{8}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}} + \frac{72x^{7}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}} - \frac{54x^{6}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}} - \frac{72x^{5}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}} - \frac{24(-x^{2} + 1)^{\frac{1}{2}}x^{3}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} + \frac{48(-x^{2} + 1)^{2}x^{4}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}} - \frac{18x^{4}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}} + \frac{24(-x^{2} + 1)^{2}x^{3}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}} - \frac{18(-x^{2} + 1)^{\frac{1}{2}}x^{3}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{9(-x^{2} + 1)^{\frac{1}{2}}x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{8x^{4}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{3}(-x^{2} + 1)} - \frac{8x^{3}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{3}(-x^{2} + 1)} - \frac{2x^{3}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{4(-x^{2} + 1)^{\frac{3}{2}}x}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{6(-x^{2} + 1)^{\frac{1}{2}}x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} + \frac{4(-x^{2} + 1)^{\frac{3}{2}}x}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{2}} - \frac{x^{2}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{96x}{(2x + 1)^{4}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{8sqrt(-x^{2} + 1)^{2}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{3}} + \frac{12x^{3}}{(-x^{2} + 1)^{\frac{5}{2}}(2x + 1)^{2}} - \frac{4x}{(-x^{2} + 1)^{\frac{1}{2}}(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}} - \frac{2x^{2}}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{3}(-x^{2} + 1)} - \frac{8(-x^{2} + 1)^{3}x^{2}}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})^{3}} - \frac{2x}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{8}{(2x + 1)^{3}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(2xsqrt(-x^{2} + 1) + sqrt(-x^{2} + 1))^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{2}{(2(-x^{2} + 1)^{\frac{3}{2}}x + (-x^{2} + 1)^{\frac{3}{2}})}\\ \end{split}\end{equation} \]

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