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。
\[ \begin{equation}\begin{split}【1/1】求函数arcsin(log_{5}^{x}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arcsin(log_{5}^{x})\right)}{dx}\\=&(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}))}{((1 - (log_{5}^{x})^{2})^{\frac{1}{2}})})\\=&\frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}xln(5)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}xln(5)}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}})}{xln(5)} + \frac{-1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{2}ln(5)} + \frac{-0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}xln^{2}(5)(5)}\\=&\frac{log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{2}ln^{2}(5)} - \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{2}ln(5)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{2}ln^{2}(5)} - \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{2}ln(5)}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}})log_{5}^{x}}{x^{2}ln^{2}(5)} + \frac{-2log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{2}(5)} + \frac{(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))})}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{2}ln^{2}(5)} + \frac{log_{5}^{x}*-2*0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{2}ln^{3}(5)(5)} - \frac{(\frac{\frac{-1}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln(5)} - \frac{-2}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{3}ln(5)} - \frac{-0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}(5)(5)}\\=&\frac{3{\left(log_{5}^{x}\right)}^{2}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{3}ln^{3}(5)} - \frac{3log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{2}(5)} + \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{3}(5)} + \frac{2}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{3}ln(5)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3{\left(log_{5}^{x}\right)}^{2}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{3}ln^{3}(5)} - \frac{3log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{2}(5)} + \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{3}(5)} + \frac{2}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{3}ln(5)}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{7}{2}}}){\left(log_{5}^{x}\right)}^{2}}{x^{3}ln^{3}(5)} + \frac{3*-3{\left(log_{5}^{x}\right)}^{2}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{4}ln^{3}(5)} + \frac{3(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))})}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{3}ln^{3}(5)} + \frac{3{\left(log_{5}^{x}\right)}^{2}*-3*0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{3}ln^{4}(5)(5)} - \frac{3(\frac{\frac{-3}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}})log_{5}^{x}}{x^{3}ln^{2}(5)} - \frac{3*-3log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{4}ln^{2}(5)} - \frac{3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))})}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{2}(5)} - \frac{3log_{5}^{x}*-2*0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{3}(5)(5)} + \frac{(\frac{\frac{-3}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}})}{x^{3}ln^{3}(5)} + \frac{-3}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{4}ln^{3}(5)} + \frac{-3*0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{4}(5)(5)} + \frac{2(\frac{\frac{-1}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}})}{x^{3}ln(5)} + \frac{2*-3}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{4}ln(5)} + \frac{2*-0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{3}ln^{2}(5)(5)}\\=&\frac{15{\left(log_{5}^{x}\right)}^{3}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{7}{2}}x^{4}ln^{4}(5)} - \frac{18{\left(log_{5}^{x}\right)}^{2}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{4}ln^{3}(5)} + \frac{9log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{4}ln^{4}(5)} + \frac{11log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{4}ln^{2}(5)} - \frac{6}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{4}ln^{3}(5)} - \frac{6}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{4}ln(5)}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数arcsin(log_{5}^{x}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arcsin(log_{5}^{x})\right)}{dx}\\=&(\frac{((\frac{(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}))}{((1 - (log_{5}^{x})^{2})^{\frac{1}{2}})})\\=&\frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}xln(5)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}xln(5)}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}})}{xln(5)} + \frac{-1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{2}ln(5)} + \frac{-0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}xln^{2}(5)(5)}\\=&\frac{log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{2}ln^{2}(5)} - \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{2}ln(5)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{2}ln^{2}(5)} - \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{2}ln(5)}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}})log_{5}^{x}}{x^{2}ln^{2}(5)} + \frac{-2log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{2}(5)} + \frac{(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))})}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{2}ln^{2}(5)} + \frac{log_{5}^{x}*-2*0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{2}ln^{3}(5)(5)} - \frac{(\frac{\frac{-1}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln(5)} - \frac{-2}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{3}ln(5)} - \frac{-0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}(5)(5)}\\=&\frac{3{\left(log_{5}^{x}\right)}^{2}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{3}ln^{3}(5)} - \frac{3log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{2}(5)} + \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{3}(5)} + \frac{2}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{3}ln(5)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3{\left(log_{5}^{x}\right)}^{2}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{3}ln^{3}(5)} - \frac{3log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{2}(5)} + \frac{1}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{3}(5)} + \frac{2}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{3}ln(5)}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{7}{2}}}){\left(log_{5}^{x}\right)}^{2}}{x^{3}ln^{3}(5)} + \frac{3*-3{\left(log_{5}^{x}\right)}^{2}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{4}ln^{3}(5)} + \frac{3(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))})}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{3}ln^{3}(5)} + \frac{3{\left(log_{5}^{x}\right)}^{2}*-3*0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{3}ln^{4}(5)(5)} - \frac{3(\frac{\frac{-3}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}})log_{5}^{x}}{x^{3}ln^{2}(5)} - \frac{3*-3log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{4}ln^{2}(5)} - \frac{3(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))})}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{2}(5)} - \frac{3log_{5}^{x}*-2*0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{3}(5)(5)} + \frac{(\frac{\frac{-3}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}})}{x^{3}ln^{3}(5)} + \frac{-3}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{4}ln^{3}(5)} + \frac{-3*0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{3}ln^{4}(5)(5)} + \frac{2(\frac{\frac{-1}{2}(-(\frac{2log_{5}^{x}(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))}) + 0)}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}})}{x^{3}ln(5)} + \frac{2*-3}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{4}ln(5)} + \frac{2*-0}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{3}ln^{2}(5)(5)}\\=&\frac{15{\left(log_{5}^{x}\right)}^{3}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{7}{2}}x^{4}ln^{4}(5)} - \frac{18{\left(log_{5}^{x}\right)}^{2}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{4}ln^{3}(5)} + \frac{9log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{5}{2}}x^{4}ln^{4}(5)} + \frac{11log_{5}^{x}}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{4}ln^{2}(5)} - \frac{6}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{3}{2}}x^{4}ln^{3}(5)} - \frac{6}{(-{\left(log_{5}^{x}\right)}^{2} + 1)^{\frac{1}{2}}x^{4}ln(5)}\\ \end{split}\end{equation} \]
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