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 2 个题目：每一题对 x 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/2】求函数log_{x}^{10} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( log_{x}^{10}\right)}{dx}\\=&(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})\\=&\frac{-log_{x}^{10}}{xln(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-log_{x}^{10}}{xln(x)}\right)}{dx}\\=&\frac{--log_{x}^{10}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{10}*-1}{xln^{2}(x)(x)}\\=&\frac{log_{x}^{10}}{x^{2}ln(x)} + \frac{2log_{x}^{10}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{log_{x}^{10}}{x^{2}ln(x)} + \frac{2log_{x}^{10}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2log_{x}^{10}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{10}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{10}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{10}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{-2log_{x}^{10}}{x^{3}ln(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-2log_{x}^{10}}{x^{3}ln(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{-2*-3log_{x}^{10}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{10}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{10}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{10}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{10}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{6log_{x}^{10}}{x^{4}ln(x)} + \frac{22log_{x}^{10}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{10}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{10}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数\frac{1}{lg(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{1}{lg(x)}\right)}{dx}\\=&\frac{-1}{lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{xln{10}lg^{2}(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-1}{xln{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--1}{x^{2}ln{10}lg^{2}(x)} - \frac{-0}{xln^{2}{10}lg^{2}(x)} - \frac{-2}{xln{10}lg^{3}(x)ln{10}(x)}\\=&\frac{1}{x^{2}ln{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}lg^{3}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{1}{x^{2}ln{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}lg^{3}(x)}\right)}{dx}\\=&\frac{-2}{x^{3}ln{10}lg^{2}(x)} + \frac{-0}{x^{2}ln^{2}{10}lg^{2}(x)} + \frac{-2}{x^{2}ln{10}lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}{10}lg^{3}(x)} + \frac{2*-2*0}{x^{2}ln^{3}{10}lg^{3}(x)} + \frac{2*-3}{x^{2}ln^{2}{10}lg^{4}(x)ln{10}(x)}\\=&\frac{-2}{x^{3}ln{10}lg^{2}(x)} - \frac{6}{x^{3}ln^{2}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{3}{10}lg^{4}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-2}{x^{3}ln{10}lg^{2}(x)} - \frac{6}{x^{3}ln^{2}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{3}{10}lg^{4}(x)}\right)}{dx}\\=&\frac{-2*-3}{x^{4}ln{10}lg^{2}(x)} - \frac{2*-0}{x^{3}ln^{2}{10}lg^{2}(x)} - \frac{2*-2}{x^{3}ln{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{2}{10}lg^{3}(x)} - \frac{6*-2*0}{x^{3}ln^{3}{10}lg^{3}(x)} - \frac{6*-3}{x^{3}ln^{2}{10}lg^{4}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{3}{10}lg^{4}(x)} - \frac{6*-3*0}{x^{3}ln^{4}{10}lg^{4}(x)} - \frac{6*-4}{x^{3}ln^{3}{10}lg^{5}(x)ln{10}(x)}\\=&\frac{6}{x^{4}ln{10}lg^{2}(x)} + \frac{22}{x^{4}ln^{2}{10}lg^{3}(x)} + \frac{36}{x^{4}ln^{3}{10}lg^{4}(x)} + \frac{24}{x^{4}ln^{4}{10}lg^{5}(x)}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/2】求函数log_{x}^{10} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( log_{x}^{10}\right)}{dx}\\=&(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})\\=&\frac{-log_{x}^{10}}{xln(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-log_{x}^{10}}{xln(x)}\right)}{dx}\\=&\frac{--log_{x}^{10}}{x^{2}ln(x)} - \frac{(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{xln(x)} - \frac{log_{x}^{10}*-1}{xln^{2}(x)(x)}\\=&\frac{log_{x}^{10}}{x^{2}ln(x)} + \frac{2log_{x}^{10}}{x^{2}ln^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{log_{x}^{10}}{x^{2}ln(x)} + \frac{2log_{x}^{10}}{x^{2}ln^{2}(x)}\right)}{dx}\\=&\frac{-2log_{x}^{10}}{x^{3}ln(x)} + \frac{(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{2}ln(x)} + \frac{log_{x}^{10}*-1}{x^{2}ln^{2}(x)(x)} + \frac{2*-2log_{x}^{10}}{x^{3}ln^{2}(x)} + \frac{2(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{2}ln^{2}(x)} + \frac{2log_{x}^{10}*-2}{x^{2}ln^{3}(x)(x)}\\=&\frac{-2log_{x}^{10}}{x^{3}ln(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{3}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-2log_{x}^{10}}{x^{3}ln(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}}{x^{3}ln^{3}(x)}\right)}{dx}\\=&\frac{-2*-3log_{x}^{10}}{x^{4}ln(x)} - \frac{2(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln(x)} - \frac{2log_{x}^{10}*-1}{x^{3}ln^{2}(x)(x)} - \frac{6*-3log_{x}^{10}}{x^{4}ln^{2}(x)} - \frac{6(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln^{2}(x)} - \frac{6log_{x}^{10}*-2}{x^{3}ln^{3}(x)(x)} - \frac{6*-3log_{x}^{10}}{x^{4}ln^{3}(x)} - \frac{6(\frac{(\frac{(0)}{(10)} - \frac{(1)log_{x}^{10}}{(x)})}{(ln(x))})}{x^{3}ln^{3}(x)} - \frac{6log_{x}^{10}*-3}{x^{3}ln^{4}(x)(x)}\\=&\frac{6log_{x}^{10}}{x^{4}ln(x)} + \frac{22log_{x}^{10}}{x^{4}ln^{2}(x)} + \frac{36log_{x}^{10}}{x^{4}ln^{3}(x)} + \frac{24log_{x}^{10}}{x^{4}ln^{4}(x)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数\frac{1}{lg(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{1}{lg(x)}\right)}{dx}\\=&\frac{-1}{lg^{2}(x)ln{10}(x)}\\=&\frac{-1}{xln{10}lg^{2}(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-1}{xln{10}lg^{2}(x)}\right)}{dx}\\=&\frac{--1}{x^{2}ln{10}lg^{2}(x)} - \frac{-0}{xln^{2}{10}lg^{2}(x)} - \frac{-2}{xln{10}lg^{3}(x)ln{10}(x)}\\=&\frac{1}{x^{2}ln{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}lg^{3}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{1}{x^{2}ln{10}lg^{2}(x)} + \frac{2}{x^{2}ln^{2}{10}lg^{3}(x)}\right)}{dx}\\=&\frac{-2}{x^{3}ln{10}lg^{2}(x)} + \frac{-0}{x^{2}ln^{2}{10}lg^{2}(x)} + \frac{-2}{x^{2}ln{10}lg^{3}(x)ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}{10}lg^{3}(x)} + \frac{2*-2*0}{x^{2}ln^{3}{10}lg^{3}(x)} + \frac{2*-3}{x^{2}ln^{2}{10}lg^{4}(x)ln{10}(x)}\\=&\frac{-2}{x^{3}ln{10}lg^{2}(x)} - \frac{6}{x^{3}ln^{2}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{3}{10}lg^{4}(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-2}{x^{3}ln{10}lg^{2}(x)} - \frac{6}{x^{3}ln^{2}{10}lg^{3}(x)} - \frac{6}{x^{3}ln^{3}{10}lg^{4}(x)}\right)}{dx}\\=&\frac{-2*-3}{x^{4}ln{10}lg^{2}(x)} - \frac{2*-0}{x^{3}ln^{2}{10}lg^{2}(x)} - \frac{2*-2}{x^{3}ln{10}lg^{3}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{2}{10}lg^{3}(x)} - \frac{6*-2*0}{x^{3}ln^{3}{10}lg^{3}(x)} - \frac{6*-3}{x^{3}ln^{2}{10}lg^{4}(x)ln{10}(x)} - \frac{6*-3}{x^{4}ln^{3}{10}lg^{4}(x)} - \frac{6*-3*0}{x^{3}ln^{4}{10}lg^{4}(x)} - \frac{6*-4}{x^{3}ln^{3}{10}lg^{5}(x)ln{10}(x)}\\=&\frac{6}{x^{4}ln{10}lg^{2}(x)} + \frac{22}{x^{4}ln^{2}{10}lg^{3}(x)} + \frac{36}{x^{4}ln^{3}{10}lg^{4}(x)} + \frac{24}{x^{4}ln^{4}{10}lg^{5}(x)}\\ \end{split}\end{equation} \]
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