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

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