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。
\[ \begin{equation}\begin{split}【1/1】求函数ln(x + {(1 + {x}^{2})}^{\frac{1}{2}}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = ln(x + (x^{2} + 1)^{\frac{1}{2}})\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( ln(x + (x^{2} + 1)^{\frac{1}{2}})\right)}{dx}\\=&\frac{(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})}\\=&\frac{x}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{1}{2}}} + \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{1}{2}}} + \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})}\right)}{dx}\\=&\frac{(\frac{-(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}})x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + (x^{2} + 1)^{\frac{1}{2}})} + \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{1}{2}}} + (\frac{-(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}})\\=& - \frac{x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)} - \frac{2x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)} - \frac{2x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}}\right)}{dx}\\=& - \frac{(\frac{-2(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}})x^{2}}{(x^{2} + 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} - \frac{2x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)} - \frac{2(\frac{-2(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}})x}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{2(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} - \frac{2}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{-(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}})x^{2}}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})} - \frac{2x}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{-(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}})}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})}{(x + (x^{2} + 1)^{\frac{1}{2}})} - (\frac{-2(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}})\\=&\frac{2x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{2}} - \frac{2x}{(x^{2} + 1)(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} + \frac{6x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)} + \frac{6x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}(x + (x^{2} + 1)^{\frac{1}{2}})} - \frac{x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)} - \frac{x}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{3}{2}}} - \frac{3}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{1}{2}}} + \frac{2}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{2}} - \frac{2x}{(x^{2} + 1)(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} + \frac{6x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)} + \frac{6x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{5}{2}}} - \frac{2x}{(x^{2} + 1)^{\frac{3}{2}}(x + (x^{2} + 1)^{\frac{1}{2}})} - \frac{x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)} - \frac{x}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{3}{2}}} - \frac{3}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{1}{2}}} + \frac{2}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}}\right)}{dx}\\=&\frac{2(\frac{-3(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}})x^{3}}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}} + \frac{2*3x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{-2(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}})x^{3}}{(x^{2} + 1)^{2}} + \frac{3(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} + \frac{3*3x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{2}} - \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} - \frac{2(\frac{-2(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}})x}{(x^{2} + 1)} - \frac{2}{(x^{2} + 1)(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} + \frac{6(\frac{-3(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}})x^{2}}{(x^{2} + 1)} + \frac{6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}} + \frac{6*2x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)} + \frac{6(\frac{-3(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}})x}{(x^{2} + 1)^{\frac{1}{2}}} + \frac{6(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}} + \frac{6}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{1}{2}}} + \frac{3(\frac{-2(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}})x^{2}}{(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} + \frac{3*2x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{3(\frac{-(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}})x^{3}}{(x^{2} + 1)^{\frac{5}{2}}} + \frac{3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 1)^{\frac{7}{2}}})x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})} + \frac{3*3x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{5}{2}}} - \frac{2(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x}{(x + (x^{2} + 1)^{\frac{1}{2}})} - \frac{2(\frac{-(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}})x}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} + 1)^{\frac{3}{2}}(x + (x^{2} + 1)^{\frac{1}{2}})} - \frac{(\frac{-2(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}})x}{(x^{2} + 1)} - \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} - \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)} - \frac{(\frac{-(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}})x}{(x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 1)^{\frac{5}{2}}})x}{(x + (x^{2} + 1)^{\frac{1}{2}})} - \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{3}{2}}} - \frac{3(\frac{-2(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}})}{(x^{2} + 1)^{\frac{1}{2}}} - \frac{3(\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{3}{2}}})}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} + 2(\frac{-3(1 + (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 1)^{\frac{1}{2}}}))}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}})\\=& - \frac{6x^{4}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}(x^{2} + 1)^{2}} - \frac{6x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{3}{2}}} - \frac{12x^{4}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{5}{2}}} + \frac{6x^{2}}{(x^{2} + 1)^{\frac{3}{2}}(x + (x^{2} + 1)^{\frac{1}{2}})^{3}} - \frac{24x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{2}} - \frac{15x^{4}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{3}} + \frac{13x^{2}}{(x^{2} + 1)^{2}(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} - \frac{24x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}(x^{2} + 1)^{\frac{3}{2}}} - \frac{36x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}(x^{2} + 1)} + \frac{12x}{(x^{2} + 1)(x + (x^{2} + 1)^{\frac{1}{2}})^{3}} - \frac{24x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}(x^{2} + 1)^{\frac{1}{2}}} + \frac{5x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{2}} - \frac{12x^{3}}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{5}{2}}} + \frac{6x}{(x^{2} + 1)^{\frac{3}{2}}(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} - \frac{15x^{4}}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{7}{2}}} + \frac{15x^{2}}{(x^{2} + 1)^{\frac{5}{2}}(x + (x^{2} + 1)^{\frac{1}{2}})} + \frac{6x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)^{\frac{3}{2}}} + \frac{12x}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)} + \frac{3x^{2}}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})^{2}(x^{2} + 1)} + \frac{12}{(x + (x^{2} + 1)^{\frac{1}{2}})^{3}(x^{2} + 1)^{\frac{1}{2}}} - \frac{2}{(x^{2} + 1)(x + (x^{2} + 1)^{\frac{1}{2}})^{2}} - \frac{1}{(x + (x^{2} + 1)^{\frac{1}{2}})(x^{2} + 1)^{\frac{3}{2}}} - \frac{2}{(x^{2} + 1)^{\frac{3}{2}}(x + (x^{2} + 1)^{\frac{1}{2}})} - \frac{6}{(x + (x^{2} + 1)^{\frac{1}{2}})^{4}}\\ \end{split}\end{equation} \]

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