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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{e^{\frac{-18.61}{x}}}{x} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{e^{\frac{-18.61}{x}}}{x}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{e^{\frac{-18.61}{x}}}{x}\right)}{dx}\\=&\frac{-e^{\frac{-18.61}{x}}}{x^{2}} + \frac{e^{\frac{-18.61}{x}}*-18.61*-1}{xx^{2}}\\=&\frac{-e^{\frac{-18.61}{x}}}{x^{2}} + \frac{18.61e^{\frac{-18.61}{x}}}{x^{3}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-e^{\frac{-18.61}{x}}}{x^{2}} + \frac{18.61e^{\frac{-18.61}{x}}}{x^{3}}\right)}{dx}\\=&\frac{-*-2e^{\frac{-18.61}{x}}}{x^{3}} - \frac{e^{\frac{-18.61}{x}}*-18.61*-1}{x^{2}x^{2}} + \frac{18.61*-3e^{\frac{-18.61}{x}}}{x^{4}} + \frac{18.61e^{\frac{-18.61}{x}}*-18.61*-1}{x^{3}x^{2}}\\=&\frac{2e^{\frac{-18.61}{x}}}{x^{3}} - \frac{18.61e^{\frac{-18.61}{x}}}{x^{4}} - \frac{55.83e^{\frac{-18.61}{x}}}{x^{4}} + \frac{346.3321e^{\frac{-18.61}{x}}}{x^{5}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2e^{\frac{-18.61}{x}}}{x^{3}} - \frac{18.61e^{\frac{-18.61}{x}}}{x^{4}} - \frac{55.83e^{\frac{-18.61}{x}}}{x^{4}} + \frac{346.3321e^{\frac{-18.61}{x}}}{x^{5}}\right)}{dx}\\=&\frac{2*-3e^{\frac{-18.61}{x}}}{x^{4}} + \frac{2e^{\frac{-18.61}{x}}*-18.61*-1}{x^{3}x^{2}} - \frac{18.61*-4e^{\frac{-18.61}{x}}}{x^{5}} - \frac{18.61e^{\frac{-18.61}{x}}*-18.61*-1}{x^{4}x^{2}} - \frac{55.83*-4e^{\frac{-18.61}{x}}}{x^{5}} - \frac{55.83e^{\frac{-18.61}{x}}*-18.61*-1}{x^{4}x^{2}} + \frac{346.3321*-5e^{\frac{-18.61}{x}}}{x^{6}} + \frac{346.3321e^{\frac{-18.61}{x}}*-18.61*-1}{x^{5}x^{2}}\\=&\frac{-6e^{\frac{-18.61}{x}}}{x^{4}} + \frac{37.22e^{\frac{-18.61}{x}}}{x^{5}} + \frac{74.44e^{\frac{-18.61}{x}}}{x^{5}} - \frac{346.3321e^{\frac{-18.61}{x}}}{x^{6}} + \frac{223.32e^{\frac{-18.61}{x}}}{x^{5}} - \frac{1038.9963e^{\frac{-18.61}{x}}}{x^{6}} - \frac{1731.6605e^{\frac{-18.61}{x}}}{x^{6}} + \frac{6445.240381e^{\frac{-18.61}{x}}}{x^{7}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-6e^{\frac{-18.61}{x}}}{x^{4}} + \frac{37.22e^{\frac{-18.61}{x}}}{x^{5}} + \frac{74.44e^{\frac{-18.61}{x}}}{x^{5}} - \frac{346.3321e^{\frac{-18.61}{x}}}{x^{6}} + \frac{223.32e^{\frac{-18.61}{x}}}{x^{5}} - \frac{1038.9963e^{\frac{-18.61}{x}}}{x^{6}} - \frac{1731.6605e^{\frac{-18.61}{x}}}{x^{6}} + \frac{6445.240381e^{\frac{-18.61}{x}}}{x^{7}}\right)}{dx}\\=&\frac{-6*-4e^{\frac{-18.61}{x}}}{x^{5}} - \frac{6e^{\frac{-18.61}{x}}*-18.61*-1}{x^{4}x^{2}} + \frac{37.22*-5e^{\frac{-18.61}{x}}}{x^{6}} + \frac{37.22e^{\frac{-18.61}{x}}*-18.61*-1}{x^{5}x^{2}} + \frac{74.44*-5e^{\frac{-18.61}{x}}}{x^{6}} + \frac{74.44e^{\frac{-18.61}{x}}*-18.61*-1}{x^{5}x^{2}} - \frac{346.3321*-6e^{\frac{-18.61}{x}}}{x^{7}} - \frac{346.3321e^{\frac{-18.61}{x}}*-18.61*-1}{x^{6}x^{2}} + \frac{223.32*-5e^{\frac{-18.61}{x}}}{x^{6}} + \frac{223.32e^{\frac{-18.61}{x}}*-18.61*-1}{x^{5}x^{2}} - \frac{1038.9963*-6e^{\frac{-18.61}{x}}}{x^{7}} - \frac{1038.9963e^{\frac{-18.61}{x}}*-18.61*-1}{x^{6}x^{2}} - \frac{1731.6605*-6e^{\frac{-18.61}{x}}}{x^{7}} - \frac{1731.6605e^{\frac{-18.61}{x}}*-18.61*-1}{x^{6}x^{2}} + \frac{6445.240381*-7e^{\frac{-18.61}{x}}}{x^{8}} + \frac{6445.240381e^{\frac{-18.61}{x}}*-18.61*-1}{x^{7}x^{2}}\\=&\frac{24e^{\frac{-18.61}{x}}}{x^{5}} - \frac{111.66e^{\frac{-18.61}{x}}}{x^{6}} - \frac{186.1e^{\frac{-18.61}{x}}}{x^{6}} + \frac{692.6642e^{\frac{-18.61}{x}}}{x^{7}} - \frac{372.2e^{\frac{-18.61}{x}}}{x^{6}} + \frac{1385.3284e^{\frac{-18.61}{x}}}{x^{7}} + \frac{2077.9926e^{\frac{-18.61}{x}}}{x^{7}} - \frac{6445.240381e^{\frac{-18.61}{x}}}{x^{8}} - \frac{1116.6e^{\frac{-18.61}{x}}}{x^{6}} + \frac{4155.9852e^{\frac{-18.61}{x}}}{x^{7}} + \frac{6233.9778e^{\frac{-18.61}{x}}}{x^{7}} - \frac{19335.721143e^{\frac{-18.61}{x}}}{x^{8}} + \frac{10389.963e^{\frac{-18.61}{x}}}{x^{7}} - \frac{32226.201905e^{\frac{-18.61}{x}}}{x^{8}} - \frac{45116.682667e^{\frac{-18.61}{x}}}{x^{8}} + \frac{119945.92349041e^{\frac{-18.61}{x}}}{x^{9}}\\ \end{split}\end{equation} \]

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