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。
\[ \begin{equation}\begin{split}【1/1】求函数xarctan(\frac{1}{x}) + \frac{arctan(x)}{x} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = xarctan(\frac{1}{x}) + \frac{arctan(x)}{x}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xarctan(\frac{1}{x}) + \frac{arctan(x)}{x}\right)}{dx}\\=&arctan(\frac{1}{x}) + x(\frac{(\frac{-1}{x^{2}})}{(1 + (\frac{1}{x})^{2})}) + \frac{-arctan(x)}{x^{2}} + \frac{(\frac{(1)}{(1 + (x)^{2})})}{x}\\=&arctan(\frac{1}{x}) - \frac{1}{(\frac{1}{x^{2}} + 1)x} - \frac{arctan(x)}{x^{2}} + \frac{1}{(x^{2} + 1)x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( arctan(\frac{1}{x}) - \frac{1}{(\frac{1}{x^{2}} + 1)x} - \frac{arctan(x)}{x^{2}} + \frac{1}{(x^{2} + 1)x}\right)}{dx}\\=&(\frac{(\frac{-1}{x^{2}})}{(1 + (\frac{1}{x})^{2})}) - \frac{(\frac{-(\frac{-2}{x^{3}} + 0)}{(\frac{1}{x^{2}} + 1)^{2}})}{x} - \frac{-1}{(\frac{1}{x^{2}} + 1)x^{2}} - \frac{-2arctan(x)}{x^{3}} - \frac{(\frac{(1)}{(1 + (x)^{2})})}{x^{2}} + \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{x} + \frac{-1}{(x^{2} + 1)x^{2}}\\=& - \frac{2}{(\frac{1}{x^{2}} + 1)^{2}x^{4}} + \frac{2arctan(x)}{x^{3}} - \frac{2}{(x^{2} + 1)x^{2}} - \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{2}{(\frac{1}{x^{2}} + 1)^{2}x^{4}} + \frac{2arctan(x)}{x^{3}} - \frac{2}{(x^{2} + 1)x^{2}} - \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=& - \frac{2(\frac{-2(\frac{-2}{x^{3}} + 0)}{(\frac{1}{x^{2}} + 1)^{3}})}{x^{4}} - \frac{2*-4}{(\frac{1}{x^{2}} + 1)^{2}x^{5}} + \frac{2*-3arctan(x)}{x^{4}} + \frac{2(\frac{(1)}{(1 + (x)^{2})})}{x^{3}} - \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{x^{2}} - \frac{2*-2}{(x^{2} + 1)x^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=& - \frac{8}{(\frac{1}{x^{2}} + 1)^{3}x^{7}} + \frac{8}{(\frac{1}{x^{2}} + 1)^{2}x^{5}} - \frac{6arctan(x)}{x^{4}} + \frac{6}{(x^{2} + 1)x^{3}} + \frac{4}{(x^{2} + 1)^{2}x} + \frac{8x}{(x^{2} + 1)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( - \frac{8}{(\frac{1}{x^{2}} + 1)^{3}x^{7}} + \frac{8}{(\frac{1}{x^{2}} + 1)^{2}x^{5}} - \frac{6arctan(x)}{x^{4}} + \frac{6}{(x^{2} + 1)x^{3}} + \frac{4}{(x^{2} + 1)^{2}x} + \frac{8x}{(x^{2} + 1)^{3}}\right)}{dx}\\=& - \frac{8(\frac{-3(\frac{-2}{x^{3}} + 0)}{(\frac{1}{x^{2}} + 1)^{4}})}{x^{7}} - \frac{8*-7}{(\frac{1}{x^{2}} + 1)^{3}x^{8}} + \frac{8(\frac{-2(\frac{-2}{x^{3}} + 0)}{(\frac{1}{x^{2}} + 1)^{3}})}{x^{5}} + \frac{8*-5}{(\frac{1}{x^{2}} + 1)^{2}x^{6}} - \frac{6*-4arctan(x)}{x^{5}} - \frac{6(\frac{(1)}{(1 + (x)^{2})})}{x^{4}} + \frac{6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{x^{3}} + \frac{6*-3}{(x^{2} + 1)x^{4}} + \frac{4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})}{x} + \frac{4*-1}{(x^{2} + 1)^{2}x^{2}} + 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x + \frac{8}{(x^{2} + 1)^{3}}\\=& - \frac{48}{(\frac{1}{x^{2}} + 1)^{4}x^{10}} + \frac{88}{(\frac{1}{x^{2}} + 1)^{3}x^{8}} - \frac{40}{(\frac{1}{x^{2}} + 1)^{2}x^{6}} + \frac{24arctan(x)}{x^{5}} - \frac{24}{(x^{2} + 1)x^{4}} - \frac{16}{(x^{2} + 1)^{2}x^{2}} - \frac{48x^{2}}{(x^{2} + 1)^{4}} - \frac{8}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数xarctan(\frac{1}{x}) + \frac{arctan(x)}{x} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = xarctan(\frac{1}{x}) + \frac{arctan(x)}{x}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( xarctan(\frac{1}{x}) + \frac{arctan(x)}{x}\right)}{dx}\\=&arctan(\frac{1}{x}) + x(\frac{(\frac{-1}{x^{2}})}{(1 + (\frac{1}{x})^{2})}) + \frac{-arctan(x)}{x^{2}} + \frac{(\frac{(1)}{(1 + (x)^{2})})}{x}\\=&arctan(\frac{1}{x}) - \frac{1}{(\frac{1}{x^{2}} + 1)x} - \frac{arctan(x)}{x^{2}} + \frac{1}{(x^{2} + 1)x}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( arctan(\frac{1}{x}) - \frac{1}{(\frac{1}{x^{2}} + 1)x} - \frac{arctan(x)}{x^{2}} + \frac{1}{(x^{2} + 1)x}\right)}{dx}\\=&(\frac{(\frac{-1}{x^{2}})}{(1 + (\frac{1}{x})^{2})}) - \frac{(\frac{-(\frac{-2}{x^{3}} + 0)}{(\frac{1}{x^{2}} + 1)^{2}})}{x} - \frac{-1}{(\frac{1}{x^{2}} + 1)x^{2}} - \frac{-2arctan(x)}{x^{3}} - \frac{(\frac{(1)}{(1 + (x)^{2})})}{x^{2}} + \frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{x} + \frac{-1}{(x^{2} + 1)x^{2}}\\=& - \frac{2}{(\frac{1}{x^{2}} + 1)^{2}x^{4}} + \frac{2arctan(x)}{x^{3}} - \frac{2}{(x^{2} + 1)x^{2}} - \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{2}{(\frac{1}{x^{2}} + 1)^{2}x^{4}} + \frac{2arctan(x)}{x^{3}} - \frac{2}{(x^{2} + 1)x^{2}} - \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=& - \frac{2(\frac{-2(\frac{-2}{x^{3}} + 0)}{(\frac{1}{x^{2}} + 1)^{3}})}{x^{4}} - \frac{2*-4}{(\frac{1}{x^{2}} + 1)^{2}x^{5}} + \frac{2*-3arctan(x)}{x^{4}} + \frac{2(\frac{(1)}{(1 + (x)^{2})})}{x^{3}} - \frac{2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{x^{2}} - \frac{2*-2}{(x^{2} + 1)x^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=& - \frac{8}{(\frac{1}{x^{2}} + 1)^{3}x^{7}} + \frac{8}{(\frac{1}{x^{2}} + 1)^{2}x^{5}} - \frac{6arctan(x)}{x^{4}} + \frac{6}{(x^{2} + 1)x^{3}} + \frac{4}{(x^{2} + 1)^{2}x} + \frac{8x}{(x^{2} + 1)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( - \frac{8}{(\frac{1}{x^{2}} + 1)^{3}x^{7}} + \frac{8}{(\frac{1}{x^{2}} + 1)^{2}x^{5}} - \frac{6arctan(x)}{x^{4}} + \frac{6}{(x^{2} + 1)x^{3}} + \frac{4}{(x^{2} + 1)^{2}x} + \frac{8x}{(x^{2} + 1)^{3}}\right)}{dx}\\=& - \frac{8(\frac{-3(\frac{-2}{x^{3}} + 0)}{(\frac{1}{x^{2}} + 1)^{4}})}{x^{7}} - \frac{8*-7}{(\frac{1}{x^{2}} + 1)^{3}x^{8}} + \frac{8(\frac{-2(\frac{-2}{x^{3}} + 0)}{(\frac{1}{x^{2}} + 1)^{3}})}{x^{5}} + \frac{8*-5}{(\frac{1}{x^{2}} + 1)^{2}x^{6}} - \frac{6*-4arctan(x)}{x^{5}} - \frac{6(\frac{(1)}{(1 + (x)^{2})})}{x^{4}} + \frac{6(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{x^{3}} + \frac{6*-3}{(x^{2} + 1)x^{4}} + \frac{4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})}{x} + \frac{4*-1}{(x^{2} + 1)^{2}x^{2}} + 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x + \frac{8}{(x^{2} + 1)^{3}}\\=& - \frac{48}{(\frac{1}{x^{2}} + 1)^{4}x^{10}} + \frac{88}{(\frac{1}{x^{2}} + 1)^{3}x^{8}} - \frac{40}{(\frac{1}{x^{2}} + 1)^{2}x^{6}} + \frac{24arctan(x)}{x^{5}} - \frac{24}{(x^{2} + 1)x^{4}} - \frac{16}{(x^{2} + 1)^{2}x^{2}} - \frac{48x^{2}}{(x^{2} + 1)^{4}} - \frac{8}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
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