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。
\[ \begin{equation}\begin{split}【1/1】求函数xln(ln(ln(ln(x)))) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xln(ln(ln(ln(x))))\right)}{dx}\\=&ln(ln(ln(ln(x)))) + \frac{x}{(ln(ln(ln(x))))(ln(ln(x)))(ln(x))(x)}\\=&\frac{1}{ln(ln(x))ln(x)ln(ln(ln(x)))} + ln(ln(ln(ln(x))))\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{ln(ln(x))ln(x)ln(ln(ln(x)))} + ln(ln(ln(ln(x))))\right)}{dx}\\=&\frac{-1}{ln^{2}(ln(x))(ln(x))(x)ln(x)ln(ln(ln(x)))} + \frac{-1}{ln(ln(x))ln^{2}(x)(x)ln(ln(ln(x)))} + \frac{-1}{ln(ln(x))ln(x)ln^{2}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} + \frac{1}{(ln(ln(ln(x))))(ln(ln(x)))(ln(x))(x)}\\=&\frac{-1}{xln^{2}(ln(x))ln^{2}(x)ln(ln(ln(x)))} - \frac{1}{xln^{2}(x)ln(ln(x))ln(ln(ln(x)))} - \frac{1}{xln^{2}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} + \frac{1}{xln(ln(x))ln(x)ln(ln(ln(x)))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-1}{xln^{2}(ln(x))ln^{2}(x)ln(ln(ln(x)))} - \frac{1}{xln^{2}(x)ln(ln(x))ln(ln(ln(x)))} - \frac{1}{xln^{2}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} + \frac{1}{xln(ln(x))ln(x)ln(ln(ln(x)))}\right)}{dx}\\=&\frac{--1}{x^{2}ln^{2}(ln(x))ln^{2}(x)ln(ln(ln(x)))} - \frac{-2}{xln^{3}(ln(x))(ln(x))(x)ln^{2}(x)ln(ln(ln(x)))} - \frac{-2}{xln^{2}(ln(x))ln^{3}(x)(x)ln(ln(ln(x)))} - \frac{-1}{xln^{2}(ln(x))ln^{2}(x)ln^{2}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} - \frac{-1}{x^{2}ln^{2}(x)ln(ln(x))ln(ln(ln(x)))} - \frac{-2}{xln^{3}(x)(x)ln(ln(x))ln(ln(ln(x)))} - \frac{-1}{xln^{2}(x)ln^{2}(ln(x))(ln(x))(x)ln(ln(ln(x)))} - \frac{-1}{xln^{2}(x)ln(ln(x))ln^{2}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} - \frac{-1}{x^{2}ln^{2}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} - \frac{-2}{xln^{3}(x)(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} - \frac{-2}{xln^{2}(x)ln^{3}(ln(x))(ln(x))(x)ln^{2}(ln(ln(x)))} - \frac{-2}{xln^{2}(x)ln^{2}(ln(x))ln^{3}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} + \frac{-1}{x^{2}ln(ln(x))ln(x)ln(ln(ln(x)))} + \frac{-1}{xln^{2}(ln(x))(ln(x))(x)ln(x)ln(ln(ln(x)))} + \frac{-1}{xln(ln(x))ln^{2}(x)(x)ln(ln(ln(x)))} + \frac{-1}{xln(ln(x))ln(x)ln^{2}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)}\\=&\frac{2}{x^{2}ln^{3}(ln(x))ln^{3}(x)ln(ln(ln(x)))} + \frac{3}{x^{2}ln^{3}(x)ln^{2}(ln(x))ln(ln(ln(x)))} + \frac{3}{x^{2}ln^{3}(x)ln^{3}(ln(x))ln^{2}(ln(ln(x)))} + \frac{2}{x^{2}ln^{3}(x)ln(ln(x))ln(ln(ln(x)))} + \frac{3}{x^{2}ln^{3}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} + \frac{2}{x^{2}ln^{3}(x)ln^{3}(ln(x))ln^{3}(ln(ln(x)))} - \frac{1}{x^{2}ln(ln(x))ln(x)ln(ln(ln(x)))}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{2}{x^{2}ln^{3}(ln(x))ln^{3}(x)ln(ln(ln(x)))} + \frac{3}{x^{2}ln^{3}(x)ln^{2}(ln(x))ln(ln(ln(x)))} + \frac{3}{x^{2}ln^{3}(x)ln^{3}(ln(x))ln^{2}(ln(ln(x)))} + \frac{2}{x^{2}ln^{3}(x)ln(ln(x))ln(ln(ln(x)))} + \frac{3}{x^{2}ln^{3}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} + \frac{2}{x^{2}ln^{3}(x)ln^{3}(ln(x))ln^{3}(ln(ln(x)))} - \frac{1}{x^{2}ln(ln(x))ln(x)ln(ln(ln(x)))}\right)}{dx}\\=&\frac{2*-2}{x^{3}ln^{3}(ln(x))ln^{3}(x)ln(ln(ln(x)))} + \frac{2*-3}{x^{2}ln^{4}(ln(x))(ln(x))(x)ln^{3}(x)ln(ln(ln(x)))} + \frac{2*-3}{x^{2}ln^{3}(ln(x))ln^{4}(x)(x)ln(ln(ln(x)))} + \frac{2*-1}{x^{2}ln^{3}(ln(x))ln^{3}(x)ln^{2}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} + \frac{3*-2}{x^{3}ln^{3}(x)ln^{2}(ln(x))ln(ln(ln(x)))} + \frac{3*-3}{x^{2}ln^{4}(x)(x)ln^{2}(ln(x))ln(ln(ln(x)))} + \frac{3*-2}{x^{2}ln^{3}(x)ln^{3}(ln(x))(ln(x))(x)ln(ln(ln(x)))} + \frac{3*-1}{x^{2}ln^{3}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} + \frac{3*-2}{x^{3}ln^{3}(x)ln^{3}(ln(x))ln^{2}(ln(ln(x)))} + \frac{3*-3}{x^{2}ln^{4}(x)(x)ln^{3}(ln(x))ln^{2}(ln(ln(x)))} + \frac{3*-3}{x^{2}ln^{3}(x)ln^{4}(ln(x))(ln(x))(x)ln^{2}(ln(ln(x)))} + \frac{3*-2}{x^{2}ln^{3}(x)ln^{3}(ln(x))ln^{3}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} + \frac{2*-2}{x^{3}ln^{3}(x)ln(ln(x))ln(ln(ln(x)))} + \frac{2*-3}{x^{2}ln^{4}(x)(x)ln(ln(x))ln(ln(ln(x)))} + \frac{2*-1}{x^{2}ln^{3}(x)ln^{2}(ln(x))(ln(x))(x)ln(ln(ln(x)))} + \frac{2*-1}{x^{2}ln^{3}(x)ln(ln(x))ln^{2}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} + \frac{3*-2}{x^{3}ln^{3}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} + \frac{3*-3}{x^{2}ln^{4}(x)(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} + \frac{3*-2}{x^{2}ln^{3}(x)ln^{3}(ln(x))(ln(x))(x)ln^{2}(ln(ln(x)))} + \frac{3*-2}{x^{2}ln^{3}(x)ln^{2}(ln(x))ln^{3}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} + \frac{2*-2}{x^{3}ln^{3}(x)ln^{3}(ln(x))ln^{3}(ln(ln(x)))} + \frac{2*-3}{x^{2}ln^{4}(x)(x)ln^{3}(ln(x))ln^{3}(ln(ln(x)))} + \frac{2*-3}{x^{2}ln^{3}(x)ln^{4}(ln(x))(ln(x))(x)ln^{3}(ln(ln(x)))} + \frac{2*-3}{x^{2}ln^{3}(x)ln^{3}(ln(x))ln^{4}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)} - \frac{-2}{x^{3}ln(ln(x))ln(x)ln(ln(ln(x)))} - \frac{-1}{x^{2}ln^{2}(ln(x))(ln(x))(x)ln(x)ln(ln(ln(x)))} - \frac{-1}{x^{2}ln(ln(x))ln^{2}(x)(x)ln(ln(ln(x)))} - \frac{-1}{x^{2}ln(ln(x))ln(x)ln^{2}(ln(ln(x)))(ln(ln(x)))(ln(x))(x)}\\=& - \frac{4}{x^{3}ln^{3}(ln(x))ln^{3}(x)ln(ln(ln(x)))} - \frac{6}{x^{3}ln^{4}(ln(x))ln^{4}(x)ln(ln(ln(x)))} - \frac{12}{x^{3}ln^{4}(x)ln^{3}(ln(x))ln(ln(ln(x)))} - \frac{11}{x^{3}ln^{4}(x)ln^{4}(ln(x))ln^{2}(ln(ln(x)))} - \frac{6}{x^{3}ln^{3}(x)ln^{2}(ln(x))ln(ln(ln(x)))} - \frac{11}{x^{3}ln^{4}(x)ln^{2}(ln(x))ln(ln(ln(x)))} - \frac{18}{x^{3}ln^{4}(x)ln^{3}(ln(x))ln^{2}(ln(ln(x)))} - \frac{6}{x^{3}ln^{3}(x)ln^{3}(ln(x))ln^{2}(ln(ln(x)))} - \frac{12}{x^{3}ln^{4}(x)ln^{4}(ln(x))ln^{3}(ln(ln(x)))} - \frac{4}{x^{3}ln^{3}(x)ln(ln(x))ln(ln(ln(x)))} - \frac{6}{x^{3}ln^{4}(x)ln(ln(x))ln(ln(ln(x)))} - \frac{11}{x^{3}ln^{4}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} - \frac{6}{x^{3}ln^{3}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))} - \frac{12}{x^{3}ln^{4}(x)ln^{3}(ln(x))ln^{3}(ln(ln(x)))} - \frac{4}{x^{3}ln^{3}(x)ln^{3}(ln(x))ln^{3}(ln(ln(x)))} - \frac{6}{x^{3}ln^{4}(x)ln^{4}(ln(x))ln^{4}(ln(ln(x)))} + \frac{2}{x^{3}ln(ln(x))ln(x)ln(ln(ln(x)))} + \frac{1}{x^{3}ln^{2}(ln(x))ln^{2}(x)ln(ln(ln(x)))} + \frac{1}{x^{3}ln^{2}(x)ln(ln(x))ln(ln(ln(x)))} + \frac{1}{x^{3}ln^{2}(x)ln^{2}(ln(x))ln^{2}(ln(ln(x)))}\\ \end{split}\end{equation} \]

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