求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
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。
\[ \begin{equation}\begin{split}【1/1】求函数log_{lg(x)}^{e^{x}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( log_{lg(x)}^{e^{x}}\right)}{dx}\\=&(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})\\=&\frac{1}{ln(lg(x))} - \frac{log_{lg(x)}^{e^{x}}}{xln{10}ln(lg(x))lg(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{ln(lg(x))} - \frac{log_{lg(x)}^{e^{x}}}{xln{10}ln(lg(x))lg(x)}\right)}{dx}\\=&\frac{-1}{ln^{2}(lg(x))(lg(x))ln{10}(x)} - \frac{-log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{xln{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-0}{xln^{2}{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-1}{xln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-1}{xln{10}ln(lg(x))lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{xln{10}ln^{2}(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{1}{xln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{e^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-1}{xln{10}ln^{2}(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{1}{xln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{e^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} - \frac{-0}{xln^{2}{10}ln^{2}(lg(x))lg(x)} - \frac{-2}{xln{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{-1}{xln{10}ln^{2}(lg(x))lg^{2}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-0}{x^{2}ln^{2}{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} - \frac{-1}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{-2}{xln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} - \frac{-0}{xln^{2}(lg(x))ln^{2}{10}lg(x)} - \frac{-1}{xln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-2*0}{x^{2}ln(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-2}{x^{2}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)}\\=&\frac{1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} + \frac{4}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} + \frac{4}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\right)}{dx}\\=&\frac{-2}{x^{3}ln{10}ln^{2}(lg(x))lg(x)} + \frac{-0}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg(x)} + \frac{-2}{x^{2}ln{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{-1}{x^{2}ln{10}ln^{2}(lg(x))lg^{2}(x)ln{10}(x)} + \frac{4*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{4*-2*0}{x^{2}ln^{3}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{4*-3}{x^{2}ln^{2}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{4*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{2*-2*0}{x^{2}ln^{2}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-0}{x^{3}ln^{2}{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}(lg(x))ln{10}lg(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} + \frac{2*-0}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg(x)} + \frac{2*-1}{x^{2}ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2*0}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} - \frac{3*-3log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-1}{x^{3}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-2*0}{x^{3}ln(lg(x))ln^{3}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-2}{x^{3}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2*-3}{x^{2}ln^{4}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{2*-2*0}{x^{2}ln^{3}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{4*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{2}(lg(x))ln^{4}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{3}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln^{3}{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln(lg(x))lg^{4}(x)ln{10}(x)}\\=&\frac{-2}{x^{3}ln{10}ln^{2}(lg(x))lg(x)} - \frac{14}{x^{3}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} - \frac{8}{x^{3}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{18}{x^{3}ln^{3}{10}ln^{4}(lg(x))lg^{3}(x)} - \frac{14}{x^{3}ln^{3}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{10}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{6}{x^{3}ln^{2}(lg(x))ln{10}lg(x)} + \frac{22log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{11log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{4}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{10}{x^{3}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{36log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{24log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{12log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{12log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{4}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{4}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{24log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{4}(lg(x))lg^{4}(x)} + \frac{18log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{18log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{3}(lg(x))lg^{4}(x)} + \frac{16log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{2}(lg(x))lg^{4}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{4}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数log_{lg(x)}^{e^{x}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( log_{lg(x)}^{e^{x}}\right)}{dx}\\=&(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})\\=&\frac{1}{ln(lg(x))} - \frac{log_{lg(x)}^{e^{x}}}{xln{10}ln(lg(x))lg(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{ln(lg(x))} - \frac{log_{lg(x)}^{e^{x}}}{xln{10}ln(lg(x))lg(x)}\right)}{dx}\\=&\frac{-1}{ln^{2}(lg(x))(lg(x))ln{10}(x)} - \frac{-log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{xln{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-0}{xln^{2}{10}ln(lg(x))lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-1}{xln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{log_{lg(x)}^{e^{x}}*-1}{xln{10}ln(lg(x))lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{xln{10}ln^{2}(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{1}{xln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{e^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-1}{xln{10}ln^{2}(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln{10}ln(lg(x))lg(x)} - \frac{1}{xln^{2}(lg(x))ln{10}lg(x)} + \frac{2log_{lg(x)}^{e^{x}}}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} - \frac{-0}{xln^{2}{10}ln^{2}(lg(x))lg(x)} - \frac{-2}{xln{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{-1}{xln{10}ln^{2}(lg(x))lg^{2}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-0}{x^{2}ln^{2}{10}ln(lg(x))lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} - \frac{-1}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{-2}{xln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} - \frac{-0}{xln^{2}(lg(x))ln^{2}{10}lg(x)} - \frac{-1}{xln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} + \frac{2*-2log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{-2log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{2}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-1}{x^{2}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-2*0}{x^{2}ln(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{log_{lg(x)}^{e^{x}}*-2}{x^{2}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)}\\=&\frac{1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} + \frac{4}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{1}{x^{2}ln{10}ln^{2}(lg(x))lg(x)} + \frac{4}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln{10}ln(lg(x))lg(x)} + \frac{2}{x^{2}ln^{2}(lg(x))ln{10}lg(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{1}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)}\right)}{dx}\\=&\frac{-2}{x^{3}ln{10}ln^{2}(lg(x))lg(x)} + \frac{-0}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg(x)} + \frac{-2}{x^{2}ln{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg(x)} + \frac{-1}{x^{2}ln{10}ln^{2}(lg(x))lg^{2}(x)ln{10}(x)} + \frac{4*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{4*-2*0}{x^{2}ln^{3}{10}ln^{3}(lg(x))lg^{2}(x)} + \frac{4*-3}{x^{2}ln^{2}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{4*-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{2*-2*0}{x^{2}ln^{2}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-0}{x^{3}ln^{2}{10}ln(lg(x))lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln{10}ln(lg(x))lg^{2}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}(lg(x))ln{10}lg(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))(lg(x))ln{10}(x)ln{10}lg(x)} + \frac{2*-0}{x^{2}ln^{2}(lg(x))ln^{2}{10}lg(x)} + \frac{2*-1}{x^{2}ln^{2}(lg(x))ln{10}lg^{2}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2*0}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} - \frac{3*-3log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-1}{x^{3}ln^{2}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-2*0}{x^{3}ln(lg(x))ln^{3}{10}lg^{2}(x)} - \frac{3log_{lg(x)}^{e^{x}}*-2}{x^{3}ln(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} + \frac{-2}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{-2*0}{x^{2}ln^{3}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{-2}{x^{2}ln^{2}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{2}(x)} + \frac{-2}{x^{2}ln^{2}{10}ln^{2}(lg(x))lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{2*-3}{x^{2}ln^{4}(lg(x))(lg(x))ln{10}(x)ln^{2}{10}lg^{2}(x)} + \frac{2*-2*0}{x^{2}ln^{3}(lg(x))ln^{3}{10}lg^{2}(x)} + \frac{2*-2}{x^{2}ln^{3}(lg(x))ln^{2}{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{4}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{6log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{4*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{3}(lg(x))(lg(x))ln{10}(x)ln^{3}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{2}(lg(x))ln^{4}{10}lg^{3}(x)} - \frac{4log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln^{2}(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-2}{x^{3}ln^{3}{10}ln^{3}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{4}(x)ln{10}(x)} - \frac{2*-3log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2(\frac{(\frac{(e^{x})}{(e^{x})} - \frac{(\frac{1}{ln{10}(x)})log_{lg(x)}^{e^{x}}}{(lg(x))})}{(ln(lg(x)))})}{x^{3}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3*0}{x^{3}ln^{4}{10}ln(lg(x))lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-1}{x^{3}ln^{3}{10}ln^{2}(lg(x))(lg(x))ln{10}(x)lg^{3}(x)} - \frac{2log_{lg(x)}^{e^{x}}*-3}{x^{3}ln^{3}{10}ln(lg(x))lg^{4}(x)ln{10}(x)}\\=&\frac{-2}{x^{3}ln{10}ln^{2}(lg(x))lg(x)} - \frac{14}{x^{3}ln^{2}{10}ln^{3}(lg(x))lg^{2}(x)} - \frac{8}{x^{3}ln^{2}(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{18}{x^{3}ln^{3}{10}ln^{4}(lg(x))lg^{3}(x)} - \frac{14}{x^{3}ln^{3}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{10}{x^{3}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} - \frac{4}{x^{3}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln{10}ln(lg(x))lg(x)} - \frac{6}{x^{3}ln^{2}(lg(x))ln{10}lg(x)} + \frac{22log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} + \frac{11log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{2}{10}lg^{2}(x)} - \frac{4}{x^{3}ln^{2}{10}ln^{2}(lg(x))lg^{2}(x)} - \frac{10}{x^{3}ln^{3}(lg(x))ln^{2}{10}lg^{2}(x)} + \frac{36log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{3}(lg(x))lg^{3}(x)} + \frac{24log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{12log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln^{2}(lg(x))lg^{3}(x)} + \frac{12log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}{10}ln(lg(x))lg^{3}(x)} - \frac{4}{x^{3}ln^{2}(lg(x))ln^{3}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{4}(lg(x))ln^{3}{10}lg^{3}(x)} + \frac{24log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{4}(lg(x))lg^{4}(x)} + \frac{18log_{lg(x)}^{e^{x}}}{x^{4}ln^{3}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{18log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{3}(lg(x))lg^{4}(x)} + \frac{16log_{lg(x)}^{e^{x}}}{x^{4}ln^{4}{10}ln^{2}(lg(x))lg^{4}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln^{2}(lg(x))ln^{4}{10}lg^{4}(x)} + \frac{6log_{lg(x)}^{e^{x}}}{x^{4}ln(lg(x))ln^{4}{10}lg^{4}(x)}\\ \end{split}\end{equation} \]
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