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。
\[ \begin{equation}\begin{split}【1/1】求函数sin(\frac{lg(x)}{2}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = sin(\frac{1}{2}lg(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(\frac{1}{2}lg(x))\right)}{dx}\\=&\frac{cos(\frac{1}{2}lg(x))*\frac{1}{2}}{ln{10}(x)}\\=&\frac{cos(\frac{1}{2}lg(x))}{2xln{10}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{cos(\frac{1}{2}lg(x))}{2xln{10}}\right)}{dx}\\=&\frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} + \frac{-0cos(\frac{1}{2}lg(x))}{2xln^{2}{10}} + \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{2xln{10}ln{10}(x)}\\=&\frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} - \frac{sin(\frac{1}{2}lg(x))}{4x^{2}ln^{2}{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} - \frac{sin(\frac{1}{2}lg(x))}{4x^{2}ln^{2}{10}}\right)}{dx}\\=&\frac{--2cos(\frac{1}{2}lg(x))}{2x^{3}ln{10}} - \frac{-0cos(\frac{1}{2}lg(x))}{2x^{2}ln^{2}{10}} - \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{2x^{2}ln{10}ln{10}(x)} - \frac{-2sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{-2*0sin(\frac{1}{2}lg(x))}{4x^{2}ln^{3}{10}} - \frac{cos(\frac{1}{2}lg(x))*\frac{1}{2}}{4x^{2}ln^{2}{10}ln{10}(x)}\\=&\frac{cos(\frac{1}{2}lg(x))}{x^{3}ln{10}} + \frac{3sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{cos(\frac{1}{2}lg(x))}{8x^{3}ln^{3}{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{cos(\frac{1}{2}lg(x))}{x^{3}ln{10}} + \frac{3sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{cos(\frac{1}{2}lg(x))}{8x^{3}ln^{3}{10}}\right)}{dx}\\=&\frac{-3cos(\frac{1}{2}lg(x))}{x^{4}ln{10}} + \frac{-0cos(\frac{1}{2}lg(x))}{x^{3}ln^{2}{10}} + \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3sin(\frac{1}{2}lg(x))}{4x^{4}ln^{2}{10}} + \frac{3*-2*0sin(\frac{1}{2}lg(x))}{4x^{3}ln^{3}{10}} + \frac{3cos(\frac{1}{2}lg(x))*\frac{1}{2}}{4x^{3}ln^{2}{10}ln{10}(x)} - \frac{-3cos(\frac{1}{2}lg(x))}{8x^{4}ln^{3}{10}} - \frac{-3*0cos(\frac{1}{2}lg(x))}{8x^{3}ln^{4}{10}} - \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{8x^{3}ln^{3}{10}ln{10}(x)}\\=&\frac{-3cos(\frac{1}{2}lg(x))}{x^{4}ln{10}} - \frac{11sin(\frac{1}{2}lg(x))}{4x^{4}ln^{2}{10}} + \frac{3cos(\frac{1}{2}lg(x))}{4x^{4}ln^{3}{10}} + \frac{sin(\frac{1}{2}lg(x))}{16x^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数sin(\frac{lg(x)}{2}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = sin(\frac{1}{2}lg(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(\frac{1}{2}lg(x))\right)}{dx}\\=&\frac{cos(\frac{1}{2}lg(x))*\frac{1}{2}}{ln{10}(x)}\\=&\frac{cos(\frac{1}{2}lg(x))}{2xln{10}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{cos(\frac{1}{2}lg(x))}{2xln{10}}\right)}{dx}\\=&\frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} + \frac{-0cos(\frac{1}{2}lg(x))}{2xln^{2}{10}} + \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{2xln{10}ln{10}(x)}\\=&\frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} - \frac{sin(\frac{1}{2}lg(x))}{4x^{2}ln^{2}{10}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-cos(\frac{1}{2}lg(x))}{2x^{2}ln{10}} - \frac{sin(\frac{1}{2}lg(x))}{4x^{2}ln^{2}{10}}\right)}{dx}\\=&\frac{--2cos(\frac{1}{2}lg(x))}{2x^{3}ln{10}} - \frac{-0cos(\frac{1}{2}lg(x))}{2x^{2}ln^{2}{10}} - \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{2x^{2}ln{10}ln{10}(x)} - \frac{-2sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{-2*0sin(\frac{1}{2}lg(x))}{4x^{2}ln^{3}{10}} - \frac{cos(\frac{1}{2}lg(x))*\frac{1}{2}}{4x^{2}ln^{2}{10}ln{10}(x)}\\=&\frac{cos(\frac{1}{2}lg(x))}{x^{3}ln{10}} + \frac{3sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{cos(\frac{1}{2}lg(x))}{8x^{3}ln^{3}{10}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{cos(\frac{1}{2}lg(x))}{x^{3}ln{10}} + \frac{3sin(\frac{1}{2}lg(x))}{4x^{3}ln^{2}{10}} - \frac{cos(\frac{1}{2}lg(x))}{8x^{3}ln^{3}{10}}\right)}{dx}\\=&\frac{-3cos(\frac{1}{2}lg(x))}{x^{4}ln{10}} + \frac{-0cos(\frac{1}{2}lg(x))}{x^{3}ln^{2}{10}} + \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{x^{3}ln{10}ln{10}(x)} + \frac{3*-3sin(\frac{1}{2}lg(x))}{4x^{4}ln^{2}{10}} + \frac{3*-2*0sin(\frac{1}{2}lg(x))}{4x^{3}ln^{3}{10}} + \frac{3cos(\frac{1}{2}lg(x))*\frac{1}{2}}{4x^{3}ln^{2}{10}ln{10}(x)} - \frac{-3cos(\frac{1}{2}lg(x))}{8x^{4}ln^{3}{10}} - \frac{-3*0cos(\frac{1}{2}lg(x))}{8x^{3}ln^{4}{10}} - \frac{-sin(\frac{1}{2}lg(x))*\frac{1}{2}}{8x^{3}ln^{3}{10}ln{10}(x)}\\=&\frac{-3cos(\frac{1}{2}lg(x))}{x^{4}ln{10}} - \frac{11sin(\frac{1}{2}lg(x))}{4x^{4}ln^{2}{10}} + \frac{3cos(\frac{1}{2}lg(x))}{4x^{4}ln^{3}{10}} + \frac{sin(\frac{1}{2}lg(x))}{16x^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]
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