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。
\[ \begin{equation}\begin{split}【1/2】求函数xlg(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xlg(x)\right)}{dx}\\=&lg(x) + \frac{x}{ln{10}(x)}\\=&lg(x) + \frac{1}{ln{10}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( lg(x) + \frac{1}{ln{10}}\right)}{dx}\\=&\frac{1}{ln{10}(x)} + \frac{-0}{ln^{2}{10}}\\=&\frac{1}{xln{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{1}{xln{10}}\right)}{dx}\\=&\frac{-1}{x^{2}ln{10}} + \frac{-0}{xln^{2}{10}}\\=&\frac{-1}{x^{2}ln{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-1}{x^{2}ln{10}}\right)}{dx}\\=&\frac{--2}{x^{3}ln{10}} - \frac{-0}{x^{2}ln^{2}{10}}\\=&\frac{2}{x^{3}ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数{x}^{lg(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {x}^{lg(x)}\right)}{dx}\\=&({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))\\=&\frac{{x}^{lg(x)}ln(x)}{xln{10}} + \frac{{x}^{lg(x)}lg(x)}{x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{{x}^{lg(x)}ln(x)}{xln{10}} + \frac{{x}^{lg(x)}lg(x)}{x}\right)}{dx}\\=&\frac{-{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{xln{10}} + \frac{{x}^{lg(x)}}{x(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0}{xln^{2}{10}} + \frac{-{x}^{lg(x)}lg(x)}{x^{2}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x} + \frac{{x}^{lg(x)}}{xln{10}(x)}\\=&\frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln^{2}(x)}{x^{2}ln^{2}{10}} - \frac{{x}^{lg(x)}lg(x)}{x^{2}} + \frac{2{x}^{lg(x)}}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln^{2}(x)}{x^{2}ln^{2}{10}} - \frac{{x}^{lg(x)}lg(x)}{x^{2}} + \frac{2{x}^{lg(x)}}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{2}}\right)}{dx}\\=&\frac{-2{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg(x)}{x^{2}(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0lg(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}ln{10}(x)} + \frac{-2{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}*-0ln(x)lg(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}lg(x)}{x^{2}ln{10}(x)} + \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}ln{10}(x)} - \frac{-2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}}{x^{2}(x)ln{10}} - \frac{{x}^{lg(x)}ln(x)*-0}{x^{2}ln^{2}{10}} + \frac{-2{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}*-2*0ln^{2}(x)}{x^{2}ln^{3}{10}} + \frac{{x}^{lg(x)}*2ln(x)}{x^{2}ln^{2}{10}(x)} - \frac{-2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{2}} - \frac{{x}^{lg(x)}}{x^{2}ln{10}(x)} + \frac{2*-2{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{x^{2}ln{10}} + \frac{2{x}^{lg(x)}*-0}{x^{2}ln^{2}{10}} + \frac{-2{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{2}(x)}{x^{2}} + \frac{{x}^{lg(x)}*2lg(x)}{x^{2}ln{10}(x)}\\=&\frac{-3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln^{3}(x)}{x^{3}ln^{3}{10}} - \frac{6{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{3{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{{x}^{lg(x)}lg^{3}(x)}{x^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln^{3}(x)}{x^{3}ln^{3}{10}} - \frac{6{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{3{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{{x}^{lg(x)}lg^{3}(x)}{x^{3}}\right)}{dx}\\=&\frac{-3*-3{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}lg(x)}{x^{3}(x)ln{10}} - \frac{3{x}^{lg(x)}ln(x)*-0lg(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}*-2*0ln^{2}(x)lg(x)}{x^{3}ln^{3}{10}} + \frac{3{x}^{lg(x)}*2ln(x)lg(x)}{x^{3}ln^{2}{10}(x)} + \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}ln{10}(x)} + \frac{-3{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{3}(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0lg^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{6*-3{x}^{lg(x)}lg(x)}{x^{4}ln{10}} + \frac{6({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}*-0lg(x)}{x^{3}ln^{2}{10}} + \frac{6{x}^{lg(x)}}{x^{3}ln{10}ln{10}(x)} - \frac{3*-3{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}*-0ln(x)lg(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}lg(x)}{x^{3}ln{10}(x)} - \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln{10}ln{10}(x)} + \frac{2*-3{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}*-0ln(x)lg^{2}(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}lg^{2}(x)}{x^{3}ln{10}(x)} + \frac{2{x}^{lg(x)}ln(x)*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}}{x^{3}(x)ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)*-2*0}{x^{3}ln^{3}{10}} + \frac{3*-3{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}*-2*0ln(x)}{x^{3}ln^{3}{10}} + \frac{3{x}^{lg(x)}}{x^{3}ln^{2}{10}(x)} + \frac{2*-3{x}^{lg(x)}ln(x)}{x^{4}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}}{x^{3}(x)ln{10}} + \frac{2{x}^{lg(x)}ln(x)*-0}{x^{3}ln^{2}{10}} - \frac{3*-3{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{2}{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}*-2*0ln^{2}(x)}{x^{3}ln^{3}{10}} - \frac{3{x}^{lg(x)}*2ln(x)}{x^{3}ln^{2}{10}(x)} + \frac{-3{x}^{lg(x)}ln^{3}(x)}{x^{4}ln^{3}{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{3}(x)}{x^{3}ln^{3}{10}} + \frac{{x}^{lg(x)}*-3*0ln^{3}(x)}{x^{3}ln^{4}{10}} + \frac{{x}^{lg(x)}*3ln^{2}(x)}{x^{3}ln^{3}{10}(x)} - \frac{6*-3{x}^{lg(x)}}{x^{4}ln{10}} - \frac{6({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{x^{3}ln{10}} - \frac{6{x}^{lg(x)}*-0}{x^{3}ln^{2}{10}} + \frac{2*-3{x}^{lg(x)}lg(x)}{x^{4}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{3}} + \frac{2{x}^{lg(x)}}{x^{3}ln{10}(x)} - \frac{3*-3{x}^{lg(x)}lg^{2}(x)}{x^{4}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{2}(x)}{x^{3}} - \frac{3{x}^{lg(x)}*2lg(x)}{x^{3}ln{10}(x)} + \frac{-3{x}^{lg(x)}lg^{3}(x)}{x^{4}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{3}(x)}{x^{3}} + \frac{{x}^{lg(x)}*3lg^{2}(x)}{x^{3}ln{10}(x)}\\=&\frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} - \frac{36{x}^{lg(x)}lg(x)}{x^{4}ln{10}} + \frac{13{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{4{x}^{lg(x)}ln^{3}(x)lg(x)}{x^{4}ln^{3}{10}} + \frac{6{x}^{lg(x)}ln^{2}(x)lg^{2}(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg^{3}(x)}{x^{4}ln{10}} + \frac{12{x}^{lg(x)}lg^{2}(x)}{x^{4}ln{10}} - \frac{12{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{3{x}^{lg(x)}ln(x)lg^{3}(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{9{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{3}{10}} - \frac{6{x}^{lg(x)}ln(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}ln^{3}(x)}{x^{4}ln^{3}{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{3}{10}} + \frac{{x}^{lg(x)}ln^{4}(x)}{x^{4}ln^{4}{10}} + \frac{22{x}^{lg(x)}}{x^{4}ln{10}} + \frac{12{x}^{lg(x)}}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}lg(x)}{x^{4}} + \frac{11{x}^{lg(x)}lg^{2}(x)}{x^{4}} - \frac{6{x}^{lg(x)}lg^{3}(x)}{x^{4}} + \frac{{x}^{lg(x)}lg^{4}(x)}{x^{4}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/2】求函数xlg(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xlg(x)\right)}{dx}\\=&lg(x) + \frac{x}{ln{10}(x)}\\=&lg(x) + \frac{1}{ln{10}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( lg(x) + \frac{1}{ln{10}}\right)}{dx}\\=&\frac{1}{ln{10}(x)} + \frac{-0}{ln^{2}{10}}\\=&\frac{1}{xln{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{1}{xln{10}}\right)}{dx}\\=&\frac{-1}{x^{2}ln{10}} + \frac{-0}{xln^{2}{10}}\\=&\frac{-1}{x^{2}ln{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-1}{x^{2}ln{10}}\right)}{dx}\\=&\frac{--2}{x^{3}ln{10}} - \frac{-0}{x^{2}ln^{2}{10}}\\=&\frac{2}{x^{3}ln{10}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数{x}^{lg(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( {x}^{lg(x)}\right)}{dx}\\=&({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))\\=&\frac{{x}^{lg(x)}ln(x)}{xln{10}} + \frac{{x}^{lg(x)}lg(x)}{x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{{x}^{lg(x)}ln(x)}{xln{10}} + \frac{{x}^{lg(x)}lg(x)}{x}\right)}{dx}\\=&\frac{-{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{xln{10}} + \frac{{x}^{lg(x)}}{x(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0}{xln^{2}{10}} + \frac{-{x}^{lg(x)}lg(x)}{x^{2}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x} + \frac{{x}^{lg(x)}}{xln{10}(x)}\\=&\frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln^{2}(x)}{x^{2}ln^{2}{10}} - \frac{{x}^{lg(x)}lg(x)}{x^{2}} + \frac{2{x}^{lg(x)}}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}ln^{2}(x)}{x^{2}ln^{2}{10}} - \frac{{x}^{lg(x)}lg(x)}{x^{2}} + \frac{2{x}^{lg(x)}}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{2}}\right)}{dx}\\=&\frac{-2{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}lg(x)}{x^{2}(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0lg(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}ln{10}(x)} + \frac{-2{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{2}ln{10}} + \frac{{x}^{lg(x)}*-0ln(x)lg(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}lg(x)}{x^{2}ln{10}(x)} + \frac{{x}^{lg(x)}ln(x)}{x^{2}ln{10}ln{10}(x)} - \frac{-2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{2}ln{10}} - \frac{{x}^{lg(x)}}{x^{2}(x)ln{10}} - \frac{{x}^{lg(x)}ln(x)*-0}{x^{2}ln^{2}{10}} + \frac{-2{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)}{x^{2}ln^{2}{10}} + \frac{{x}^{lg(x)}*-2*0ln^{2}(x)}{x^{2}ln^{3}{10}} + \frac{{x}^{lg(x)}*2ln(x)}{x^{2}ln^{2}{10}(x)} - \frac{-2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{2}} - \frac{{x}^{lg(x)}}{x^{2}ln{10}(x)} + \frac{2*-2{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{x^{2}ln{10}} + \frac{2{x}^{lg(x)}*-0}{x^{2}ln^{2}{10}} + \frac{-2{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{2}(x)}{x^{2}} + \frac{{x}^{lg(x)}*2lg(x)}{x^{2}ln{10}(x)}\\=&\frac{-3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln^{3}(x)}{x^{3}ln^{3}{10}} - \frac{6{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{3{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{{x}^{lg(x)}lg^{3}(x)}{x^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln(x)lg(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}ln(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln^{3}(x)}{x^{3}ln^{3}{10}} - \frac{6{x}^{lg(x)}}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}lg(x)}{x^{3}} - \frac{3{x}^{lg(x)}lg^{2}(x)}{x^{3}} + \frac{{x}^{lg(x)}lg^{3}(x)}{x^{3}}\right)}{dx}\\=&\frac{-3*-3{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}lg(x)}{x^{3}(x)ln{10}} - \frac{3{x}^{lg(x)}ln(x)*-0lg(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)lg(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}*-2*0ln^{2}(x)lg(x)}{x^{3}ln^{3}{10}} + \frac{3{x}^{lg(x)}*2ln(x)lg(x)}{x^{3}ln^{2}{10}(x)} + \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{3}ln^{2}{10}ln{10}(x)} + \frac{-3{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{{x}^{lg(x)}lg^{2}(x)}{x^{3}(x)ln{10}} + \frac{{x}^{lg(x)}ln(x)*-0lg^{2}(x)}{x^{3}ln^{2}{10}} + \frac{{x}^{lg(x)}ln(x)*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{6*-3{x}^{lg(x)}lg(x)}{x^{4}ln{10}} + \frac{6({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{3}ln{10}} + \frac{6{x}^{lg(x)}*-0lg(x)}{x^{3}ln^{2}{10}} + \frac{6{x}^{lg(x)}}{x^{3}ln{10}ln{10}(x)} - \frac{3*-3{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg(x)}{x^{3}ln{10}} - \frac{3{x}^{lg(x)}*-0ln(x)lg(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}lg(x)}{x^{3}ln{10}(x)} - \frac{3{x}^{lg(x)}ln(x)}{x^{3}ln{10}ln{10}(x)} + \frac{2*-3{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)lg^{2}(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}*-0ln(x)lg^{2}(x)}{x^{3}ln^{2}{10}} + \frac{2{x}^{lg(x)}lg^{2}(x)}{x^{3}ln{10}(x)} + \frac{2{x}^{lg(x)}ln(x)*2lg(x)}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}}{x^{3}(x)ln^{2}{10}} + \frac{3{x}^{lg(x)}ln(x)*-2*0}{x^{3}ln^{3}{10}} + \frac{3*-3{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln^{2}{10}} + \frac{3{x}^{lg(x)}*-2*0ln(x)}{x^{3}ln^{3}{10}} + \frac{3{x}^{lg(x)}}{x^{3}ln^{2}{10}(x)} + \frac{2*-3{x}^{lg(x)}ln(x)}{x^{4}ln{10}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln(x)}{x^{3}ln{10}} + \frac{2{x}^{lg(x)}}{x^{3}(x)ln{10}} + \frac{2{x}^{lg(x)}ln(x)*-0}{x^{3}ln^{2}{10}} - \frac{3*-3{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{2}{10}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{2}(x)}{x^{3}ln^{2}{10}} - \frac{3{x}^{lg(x)}*-2*0ln^{2}(x)}{x^{3}ln^{3}{10}} - \frac{3{x}^{lg(x)}*2ln(x)}{x^{3}ln^{2}{10}(x)} + \frac{-3{x}^{lg(x)}ln^{3}(x)}{x^{4}ln^{3}{10}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))ln^{3}(x)}{x^{3}ln^{3}{10}} + \frac{{x}^{lg(x)}*-3*0ln^{3}(x)}{x^{3}ln^{4}{10}} + \frac{{x}^{lg(x)}*3ln^{2}(x)}{x^{3}ln^{3}{10}(x)} - \frac{6*-3{x}^{lg(x)}}{x^{4}ln{10}} - \frac{6({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{x^{3}ln{10}} - \frac{6{x}^{lg(x)}*-0}{x^{3}ln^{2}{10}} + \frac{2*-3{x}^{lg(x)}lg(x)}{x^{4}} + \frac{2({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg(x)}{x^{3}} + \frac{2{x}^{lg(x)}}{x^{3}ln{10}(x)} - \frac{3*-3{x}^{lg(x)}lg^{2}(x)}{x^{4}} - \frac{3({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{2}(x)}{x^{3}} - \frac{3{x}^{lg(x)}*2lg(x)}{x^{3}ln{10}(x)} + \frac{-3{x}^{lg(x)}lg^{3}(x)}{x^{4}} + \frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))lg^{3}(x)}{x^{3}} + \frac{{x}^{lg(x)}*3lg^{2}(x)}{x^{3}ln{10}(x)}\\=&\frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln^{2}(x)lg(x)}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} - \frac{36{x}^{lg(x)}lg(x)}{x^{4}ln{10}} + \frac{13{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{4{x}^{lg(x)}ln^{3}(x)lg(x)}{x^{4}ln^{3}{10}} + \frac{6{x}^{lg(x)}ln^{2}(x)lg^{2}(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln(x)lg(x)}{x^{4}ln{10}} + \frac{{x}^{lg(x)}ln(x)lg^{3}(x)}{x^{4}ln{10}} + \frac{12{x}^{lg(x)}lg^{2}(x)}{x^{4}ln{10}} - \frac{12{x}^{lg(x)}ln(x)lg^{2}(x)}{x^{4}ln{10}} + \frac{3{x}^{lg(x)}ln(x)lg^{3}(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{9{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{3}{10}} - \frac{6{x}^{lg(x)}ln(x)}{x^{4}ln{10}} - \frac{18{x}^{lg(x)}ln(x)}{x^{4}ln^{2}{10}} + \frac{11{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}ln^{3}(x)}{x^{4}ln^{3}{10}} + \frac{3{x}^{lg(x)}ln^{2}(x)}{x^{4}ln^{3}{10}} + \frac{{x}^{lg(x)}ln^{4}(x)}{x^{4}ln^{4}{10}} + \frac{22{x}^{lg(x)}}{x^{4}ln{10}} + \frac{12{x}^{lg(x)}}{x^{4}ln^{2}{10}} - \frac{6{x}^{lg(x)}lg(x)}{x^{4}} + \frac{11{x}^{lg(x)}lg^{2}(x)}{x^{4}} - \frac{6{x}^{lg(x)}lg^{3}(x)}{x^{4}} + \frac{{x}^{lg(x)}lg^{4}(x)}{x^{4}}\\ \end{split}\end{equation} \]
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