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。
\[ \begin{equation}\begin{split}【1/1】求函数e^{\frac{xxx}{ln(x)}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = e^{\frac{x^{3}}{ln(x)}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{\frac{x^{3}}{ln(x)}}\right)}{dx}\\=&e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})\\=&\frac{3x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln(x)} - \frac{x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{3x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln(x)} - \frac{x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)}\right)}{dx}\\=&\frac{3*2xe^{\frac{x^{3}}{ln(x)}}}{ln(x)} + \frac{3x^{2}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln(x)} + \frac{3x^{2}e^{\frac{x^{3}}{ln(x)}}*-1}{ln^{2}(x)(x)} - \frac{2xe^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} - \frac{x^{2}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{2}(x)} - \frac{x^{2}e^{\frac{x^{3}}{ln(x)}}*-2}{ln^{3}(x)(x)}\\=&\frac{6xe^{\frac{x^{3}}{ln(x)}}}{ln(x)} + \frac{9x^{4}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} - \frac{6x^{4}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{5xe^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} + \frac{x^{4}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{2xe^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{6xe^{\frac{x^{3}}{ln(x)}}}{ln(x)} + \frac{9x^{4}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} - \frac{6x^{4}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{5xe^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} + \frac{x^{4}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{2xe^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)}\right)}{dx}\\=&\frac{6e^{\frac{x^{3}}{ln(x)}}}{ln(x)} + \frac{6xe^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln(x)} + \frac{6xe^{\frac{x^{3}}{ln(x)}}*-1}{ln^{2}(x)(x)} + \frac{9*4x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} + \frac{9x^{4}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{2}(x)} + \frac{9x^{4}e^{\frac{x^{3}}{ln(x)}}*-2}{ln^{3}(x)(x)} - \frac{6*4x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{6x^{4}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{3}(x)} - \frac{6x^{4}e^{\frac{x^{3}}{ln(x)}}*-3}{ln^{4}(x)(x)} - \frac{5e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} - \frac{5xe^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{2}(x)} - \frac{5xe^{\frac{x^{3}}{ln(x)}}*-2}{ln^{3}(x)(x)} + \frac{4x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{x^{4}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{4}(x)} + \frac{x^{4}e^{\frac{x^{3}}{ln(x)}}*-4}{ln^{5}(x)(x)} + \frac{2e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} + \frac{2xe^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{3}(x)} + \frac{2xe^{\frac{x^{3}}{ln(x)}}*-3}{ln^{4}(x)(x)}\\=&\frac{6e^{\frac{x^{3}}{ln(x)}}}{ln(x)} + \frac{54x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} - \frac{63x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{11e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} + \frac{27x^{6}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{27x^{6}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{9x^{6}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} + \frac{33x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{12e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{x^{6}e^{\frac{x^{3}}{ln(x)}}}{ln^{6}(x)} - \frac{6x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} - \frac{6e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{6e^{\frac{x^{3}}{ln(x)}}}{ln(x)} + \frac{54x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} - \frac{63x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{11e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} + \frac{27x^{6}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{27x^{6}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{9x^{6}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} + \frac{33x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{12e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{x^{6}e^{\frac{x^{3}}{ln(x)}}}{ln^{6}(x)} - \frac{6x^{3}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} - \frac{6e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)}\right)}{dx}\\=&\frac{6e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln(x)} + \frac{6e^{\frac{x^{3}}{ln(x)}}*-1}{ln^{2}(x)(x)} + \frac{54*3x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} + \frac{54x^{3}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{2}(x)} + \frac{54x^{3}e^{\frac{x^{3}}{ln(x)}}*-2}{ln^{3}(x)(x)} - \frac{63*3x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{63x^{3}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{3}(x)} - \frac{63x^{3}e^{\frac{x^{3}}{ln(x)}}*-3}{ln^{4}(x)(x)} - \frac{11e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{2}(x)} - \frac{11e^{\frac{x^{3}}{ln(x)}}*-2}{ln^{3}(x)(x)} + \frac{27*6x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} + \frac{27x^{6}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{3}(x)} + \frac{27x^{6}e^{\frac{x^{3}}{ln(x)}}*-3}{ln^{4}(x)(x)} - \frac{27*6x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} - \frac{27x^{6}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{4}(x)} - \frac{27x^{6}e^{\frac{x^{3}}{ln(x)}}*-4}{ln^{5}(x)(x)} + \frac{9*6x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} + \frac{9x^{6}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{5}(x)} + \frac{9x^{6}e^{\frac{x^{3}}{ln(x)}}*-5}{ln^{6}(x)(x)} + \frac{33*3x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{33x^{3}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{4}(x)} + \frac{33x^{3}e^{\frac{x^{3}}{ln(x)}}*-4}{ln^{5}(x)(x)} + \frac{12e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{3}(x)} + \frac{12e^{\frac{x^{3}}{ln(x)}}*-3}{ln^{4}(x)(x)} - \frac{6x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{6}(x)} - \frac{x^{6}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{6}(x)} - \frac{x^{6}e^{\frac{x^{3}}{ln(x)}}*-6}{ln^{7}(x)(x)} - \frac{6*3x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} - \frac{6x^{3}e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{5}(x)} - \frac{6x^{3}e^{\frac{x^{3}}{ln(x)}}*-5}{ln^{6}(x)(x)} - \frac{6e^{\frac{x^{3}}{ln(x)}}(\frac{3x^{2}}{ln(x)} + \frac{x^{3}*-1}{ln^{2}(x)(x)})}{ln^{4}(x)} - \frac{6e^{\frac{x^{3}}{ln(x)}}*-4}{ln^{5}(x)(x)}\\=&\frac{180x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{2}(x)} - \frac{336x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{6e^{\frac{x^{3}}{ln(x)}}}{xln^{2}(x)} + \frac{324x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{3}(x)} - \frac{486x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{324x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} + \frac{335x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} + \frac{22e^{\frac{x^{3}}{ln(x)}}}{xln^{3}(x)} + \frac{81x^{8}e^{\frac{x^{3}}{ln(x)}}}{ln^{4}(x)} - \frac{108x^{8}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} + \frac{54x^{8}e^{\frac{x^{3}}{ln(x)}}}{ln^{6}(x)} - \frac{12x^{8}e^{\frac{x^{3}}{ln(x)}}}{ln^{7}(x)} - \frac{102x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{6}(x)} - \frac{180x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{5}(x)} - \frac{36e^{\frac{x^{3}}{ln(x)}}}{xln^{4}(x)} + \frac{x^{8}e^{\frac{x^{3}}{ln(x)}}}{ln^{8}(x)} + \frac{12x^{5}e^{\frac{x^{3}}{ln(x)}}}{ln^{7}(x)} + \frac{36x^{2}e^{\frac{x^{3}}{ln(x)}}}{ln^{6}(x)} + \frac{24e^{\frac{x^{3}}{ln(x)}}}{xln^{5}(x)}\\ \end{split}\end{equation} \]

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