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。
\[ \begin{equation}\begin{split}【1/1】求函数e^{\frac{x}{(e^{x + 1})}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = e^{\frac{x}{e^{x + 1}}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{\frac{x}{e^{x + 1}}}\right)}{dx}\\=&e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})\\=&\frac{e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{x + 1}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{x + 1}}\right)}{dx}\\=&\frac{e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{x + 1}} + \frac{e^{\frac{x}{e^{x + 1}}}*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}} - \frac{e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} - \frac{x*-e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{x + 1}}\\=&\frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{2e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} + \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{2e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} + \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}}\right)}{dx}\\=&\frac{e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}} + \frac{e^{\frac{x}{e^{x + 1}}}*-2e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{3}}} - \frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{x*-2e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}} - \frac{2*-e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{2e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{x + 1}} + \frac{e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{x + 1}} - \frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}} - \frac{xe^{\frac{x}{e^{x + 1}}}*-2e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{3}}} + \frac{2xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{x^{2}*-2e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{x^{2}e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}}\\=&\frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{3e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{8xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{2xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{2x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{3e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} - \frac{3e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} + \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{3x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{x^{3}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{3e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{8xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{2xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{2x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{3e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} - \frac{3e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} + \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{3x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{x^{3}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}}\right)}{dx}\\=&\frac{e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{3}}} + \frac{e^{\frac{x}{e^{x + 1}}}*-3e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{4}}} - \frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{x*-3e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} - \frac{xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{3}}} - \frac{3*-2e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{3e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}} + \frac{8e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{8x*-2e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{8xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}} - \frac{2e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{2xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{3}}} - \frac{2xe^{\frac{x}{e^{x + 1}}}*-3e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{4}}} + \frac{2*2xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{2x^{2}*-3e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} + \frac{2x^{2}e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{3}}} + \frac{3*-e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{3e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{x + 1}} - \frac{3e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}} - \frac{3e^{\frac{x}{e^{x + 1}}}*-2e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{3}}} - \frac{e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} - \frac{x*-e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{x + 1}} + \frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{xe^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}} + \frac{xe^{\frac{x}{e^{x + 1}}}*-2e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{3}}} - \frac{3*2xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{3x^{2}*-2e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{3x^{2}e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{2}}} + \frac{2xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{x^{2}e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{3}}} + \frac{x^{2}e^{\frac{x}{e^{x + 1}}}*-3e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{4}}} - \frac{3x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{x^{3}*-3e^{x + 1}(1 + 0)e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} - \frac{x^{3}e^{\frac{x}{e^{x + 1}}}(\frac{1}{e^{x + 1}} + \frac{x*-e^{x + 1}(1 + 0)}{e^{{\left(x + 1\right)}*{2}}})}{e^{{\left(x + 1\right)}*{3}}}\\=&\frac{e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} - \frac{4e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{19xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{3xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} + \frac{3x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} + \frac{20e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{8e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{27xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{11xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{21x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{3x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} - \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{3x^{3}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} - \frac{4e^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} + \frac{4e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} + \frac{xe^{\frac{x}{e^{x + 1}}}}{e^{x + 1}} + \frac{7x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{2}}} - \frac{3x^{2}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} - \frac{x^{3}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}} + \frac{6x^{3}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{3}}} + \frac{x^{4}e^{\frac{x}{e^{x + 1}}}}{e^{{\left(x + 1\right)}*{4}}}\\ \end{split}\end{equation} \]

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