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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(-x + {(4{x}^{2} + 1)}^{\frac{1}{2}})}{({x}^{2} + 1)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{-x}{(x^{2} + 1)} + \frac{(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{-x}{(x^{2} + 1)} + \frac{(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)}\right)}{dx}\\=&-(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x - \frac{1}{(x^{2} + 1)} + (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})(4x^{2} + 1)^{\frac{1}{2}} + \frac{(\frac{\frac{1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{1}{2}}})}{(x^{2} + 1)}\\=&\frac{2x^{2}}{(x^{2} + 1)^{2}} - \frac{2(4x^{2} + 1)^{\frac{1}{2}}x}{(x^{2} + 1)^{2}} + \frac{4x}{(x^{2} + 1)(4x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{2x^{2}}{(x^{2} + 1)^{2}} - \frac{2(4x^{2} + 1)^{\frac{1}{2}}x}{(x^{2} + 1)^{2}} + \frac{4x}{(x^{2} + 1)(4x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{2} + \frac{2*2x}{(x^{2} + 1)^{2}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})(4x^{2} + 1)^{\frac{1}{2}}x - \frac{2(\frac{\frac{1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{1}{2}}})x}{(x^{2} + 1)^{2}} - \frac{2(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)^{2}} + \frac{4(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(4x^{2} + 1)^{\frac{1}{2}}} + \frac{4(\frac{\frac{-1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{3}{2}}})x}{(x^{2} + 1)} + \frac{4}{(x^{2} + 1)(4x^{2} + 1)^{\frac{1}{2}}} - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&\frac{-8x^{3}}{(x^{2} + 1)^{3}} + \frac{6x}{(x^{2} + 1)^{2}} + \frac{8(4x^{2} + 1)^{\frac{1}{2}}x^{2}}{(x^{2} + 1)^{3}} - \frac{16x^{2}}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{1}{2}}} - \frac{16x^{2}}{(x^{2} + 1)(4x^{2} + 1)^{\frac{3}{2}}} - \frac{2(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)^{2}} + \frac{4}{(x^{2} + 1)(4x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-8x^{3}}{(x^{2} + 1)^{3}} + \frac{6x}{(x^{2} + 1)^{2}} + \frac{8(4x^{2} + 1)^{\frac{1}{2}}x^{2}}{(x^{2} + 1)^{3}} - \frac{16x^{2}}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{1}{2}}} - \frac{16x^{2}}{(x^{2} + 1)(4x^{2} + 1)^{\frac{3}{2}}} - \frac{2(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)^{2}} + \frac{4}{(x^{2} + 1)(4x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3} - \frac{8*3x^{2}}{(x^{2} + 1)^{3}} + 6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{6}{(x^{2} + 1)^{2}} + 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})(4x^{2} + 1)^{\frac{1}{2}}x^{2} + \frac{8(\frac{\frac{1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{1}{2}}})x^{2}}{(x^{2} + 1)^{3}} + \frac{8(4x^{2} + 1)^{\frac{1}{2}}*2x}{(x^{2} + 1)^{3}} - \frac{16(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})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}}{(x^{2} + 1)^{2}} - \frac{16*2x}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{1}{2}}} - \frac{16(\frac{-(2x + 0)}{(x^{2} + 1)^{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}}{(x^{2} + 1)} - \frac{16*2x}{(x^{2} + 1)(4x^{2} + 1)^{\frac{3}{2}}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})(4x^{2} + 1)^{\frac{1}{2}} - \frac{2(\frac{\frac{1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{1}{2}}})}{(x^{2} + 1)^{2}} + \frac{4(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{(4x^{2} + 1)^{\frac{1}{2}}} + \frac{4(\frac{\frac{-1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{3}{2}}})}{(x^{2} + 1)}\\=&\frac{48x^{4}}{(x^{2} + 1)^{4}} - \frac{48x^{2}}{(x^{2} + 1)^{3}} - \frac{48(4x^{2} + 1)^{\frac{1}{2}}x^{3}}{(x^{2} + 1)^{4}} + \frac{96x^{3}}{(x^{2} + 1)^{3}(4x^{2} + 1)^{\frac{1}{2}}} + \frac{16(4x^{2} + 1)^{\frac{1}{2}}x}{(x^{2} + 1)^{3}} + \frac{96x^{3}}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{3}{2}}} - \frac{32x}{(4x^{2} + 1)^{\frac{1}{2}}(x^{2} + 1)^{2}} + \frac{192x^{3}}{(x^{2} + 1)(4x^{2} + 1)^{\frac{5}{2}}} - \frac{32x}{(4x^{2} + 1)^{\frac{3}{2}}(x^{2} + 1)} + \frac{8(4x^{2} + 1)^{\frac{1}{2}}x}{(x^{2} + 1)^{3}} - \frac{16x}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{1}{2}}} - \frac{16x}{(x^{2} + 1)(4x^{2} + 1)^{\frac{3}{2}}} + \frac{6}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{48x^{4}}{(x^{2} + 1)^{4}} - \frac{48x^{2}}{(x^{2} + 1)^{3}} - \frac{48(4x^{2} + 1)^{\frac{1}{2}}x^{3}}{(x^{2} + 1)^{4}} + \frac{96x^{3}}{(x^{2} + 1)^{3}(4x^{2} + 1)^{\frac{1}{2}}} + \frac{16(4x^{2} + 1)^{\frac{1}{2}}x}{(x^{2} + 1)^{3}} + \frac{96x^{3}}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{3}{2}}} - \frac{32x}{(4x^{2} + 1)^{\frac{1}{2}}(x^{2} + 1)^{2}} + \frac{192x^{3}}{(x^{2} + 1)(4x^{2} + 1)^{\frac{5}{2}}} - \frac{32x}{(4x^{2} + 1)^{\frac{3}{2}}(x^{2} + 1)} + \frac{8(4x^{2} + 1)^{\frac{1}{2}}x}{(x^{2} + 1)^{3}} - \frac{16x}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{1}{2}}} - \frac{16x}{(x^{2} + 1)(4x^{2} + 1)^{\frac{3}{2}}} + \frac{6}{(x^{2} + 1)^{2}}\right)}{dx}\\=&48(\frac{-4(2x + 0)}{(x^{2} + 1)^{5}})x^{4} + \frac{48*4x^{3}}{(x^{2} + 1)^{4}} - 48(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{48*2x}{(x^{2} + 1)^{3}} - 48(\frac{-4(2x + 0)}{(x^{2} + 1)^{5}})(4x^{2} + 1)^{\frac{1}{2}}x^{3} - \frac{48(\frac{\frac{1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{1}{2}}})x^{3}}{(x^{2} + 1)^{4}} - \frac{48(4x^{2} + 1)^{\frac{1}{2}}*3x^{2}}{(x^{2} + 1)^{4}} + \frac{96(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{3}}{(4x^{2} + 1)^{\frac{1}{2}}} + \frac{96(\frac{\frac{-1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{3}{2}}})x^{3}}{(x^{2} + 1)^{3}} + \frac{96*3x^{2}}{(x^{2} + 1)^{3}(4x^{2} + 1)^{\frac{1}{2}}} + \frac{16(\frac{\frac{1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{1}{2}}})x}{(x^{2} + 1)^{3}} + 16(4x^{2} + 1)^{\frac{1}{2}}(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x + \frac{16(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)^{3}} + \frac{96(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}}{(4x^{2} + 1)^{\frac{3}{2}}} + \frac{96(\frac{\frac{-3}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{5}{2}}})x^{3}}{(x^{2} + 1)^{2}} + \frac{96*3x^{2}}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{3}{2}}} - \frac{32(\frac{\frac{-1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{3}{2}}})x}{(x^{2} + 1)^{2}} - \frac{32(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x}{(4x^{2} + 1)^{\frac{1}{2}}} - \frac{32}{(4x^{2} + 1)^{\frac{1}{2}}(x^{2} + 1)^{2}} + \frac{192(\frac{-(2x + 0)}{(x^{2} + 1)^{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}}{(x^{2} + 1)} + \frac{192*3x^{2}}{(x^{2} + 1)(4x^{2} + 1)^{\frac{5}{2}}} - \frac{32(\frac{\frac{-3}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{5}{2}}})x}{(x^{2} + 1)} - \frac{32(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(4x^{2} + 1)^{\frac{3}{2}}} - \frac{32}{(4x^{2} + 1)^{\frac{3}{2}}(x^{2} + 1)} + 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})(4x^{2} + 1)^{\frac{1}{2}}x + \frac{8(\frac{\frac{1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{1}{2}}})x}{(x^{2} + 1)^{3}} + \frac{8(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)^{3}} - \frac{16(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x}{(4x^{2} + 1)^{\frac{1}{2}}} - \frac{16(\frac{\frac{-1}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{3}{2}}})x}{(x^{2} + 1)^{2}} - \frac{16}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{1}{2}}} - \frac{16(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(4x^{2} + 1)^{\frac{3}{2}}} - \frac{16(\frac{\frac{-3}{2}(4*2x + 0)}{(4x^{2} + 1)^{\frac{5}{2}}})x}{(x^{2} + 1)} - \frac{16}{(x^{2} + 1)(4x^{2} + 1)^{\frac{3}{2}}} + 6(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&\frac{-384x^{5}}{(x^{2} + 1)^{5}} + \frac{480x^{3}}{(x^{2} + 1)^{4}} - \frac{120x}{(x^{2} + 1)^{3}} + \frac{384(4x^{2} + 1)^{\frac{1}{2}}x^{4}}{(x^{2} + 1)^{5}} - \frac{768x^{4}}{(x^{2} + 1)^{4}(4x^{2} + 1)^{\frac{1}{2}}} - \frac{240(4x^{2} + 1)^{\frac{1}{2}}x^{2}}{(x^{2} + 1)^{4}} - \frac{768x^{4}}{(x^{2} + 1)^{3}(4x^{2} + 1)^{\frac{3}{2}}} + \frac{480x^{2}}{(4x^{2} + 1)^{\frac{1}{2}}(x^{2} + 1)^{3}} - \frac{48(4x^{2} + 1)^{\frac{1}{2}}x^{2}}{(x^{2} + 1)^{4}} - \frac{1536x^{4}}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{5}{2}}} + \frac{480x^{2}}{(4x^{2} + 1)^{\frac{3}{2}}(x^{2} + 1)^{2}} + \frac{96x^{2}}{(x^{2} + 1)^{3}(4x^{2} + 1)^{\frac{1}{2}}} - \frac{3840x^{4}}{(x^{2} + 1)(4x^{2} + 1)^{\frac{7}{2}}} + \frac{960x^{2}}{(4x^{2} + 1)^{\frac{5}{2}}(x^{2} + 1)} + \frac{96x^{2}}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{3}{2}}} + \frac{192x^{2}}{(x^{2} + 1)(4x^{2} + 1)^{\frac{5}{2}}} + \frac{16(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)^{3}} - \frac{32}{(4x^{2} + 1)^{\frac{1}{2}}(x^{2} + 1)^{2}} - \frac{16}{(x^{2} + 1)^{2}(4x^{2} + 1)^{\frac{1}{2}}} + \frac{8(4x^{2} + 1)^{\frac{1}{2}}}{(x^{2} + 1)^{3}} - \frac{32}{(4x^{2} + 1)^{\frac{3}{2}}(x^{2} + 1)} - \frac{16}{(x^{2} + 1)(4x^{2} + 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]

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