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。
\[ \begin{equation}\begin{split}【1/1】求函数arctan(arctan(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arctan(arctan(x))\right)}{dx}\\=&(\frac{((\frac{(1)}{(1 + (x)^{2})}))}{(1 + (arctan(x))^{2})})\\=&\frac{1}{(x^{2} + 1)(arctan^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{(x^{2} + 1)(arctan^{2}(x) + 1)}\right)}{dx}\\=&\frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{(arctan^{2}(x) + 1)} + \frac{(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})}{(x^{2} + 1)}\\=&\frac{-2x}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2x}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\right)}{dx}\\=&\frac{-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x}{(arctan^{2}(x) + 1)} - \frac{2(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})x}{(x^{2} + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})arctan(x)}{(x^{2} + 1)^{2}} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})arctan(x)}{(arctan^{2}(x) + 1)^{2}} - \frac{2(\frac{(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\\=&\frac{12xarctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8x^{2}}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{12xarctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8x^{2}}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)}\right)}{dx}\\=&\frac{12(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})xarctan(x)}{(x^{2} + 1)^{3}} + \frac{12(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})xarctan(x)}{(arctan^{2}(x) + 1)^{2}} + \frac{12arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{12x(\frac{(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}}{(arctan^{2}(x) + 1)} + \frac{8(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})x^{2}}{(x^{2} + 1)^{3}} + \frac{8*2x}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8(\frac{-3((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{4}})arctan^{2}(x)}{(x^{2} + 1)^{3}} + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}} + \frac{8(\frac{2arctan(x)(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})}{(arctan^{2}(x) + 1)^{2}} - \frac{2(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})}{(x^{2} + 1)^{3}} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})}{(arctan^{2}(x) + 1)} - \frac{2(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})}{(x^{2} + 1)^{2}}\\=& - \frac{96xarctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{4}} - \frac{88x^{2}arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{4}} + \frac{16arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{24x}{(x^{2} + 1)^{4}(arctan^{2}(x) + 1)^{2}} - \frac{48x^{3}}{(x^{2} + 1)^{4}(arctan^{2}(x) + 1)} + \frac{16x}{(arctan^{2}(x) + 1)(x^{2} + 1)^{3}} - \frac{48arctan^{3}(x)}{(arctan^{2}(x) + 1)^{4}(x^{2} + 1)^{4}} + \frac{24arctan(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{4}} + \frac{8x}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数arctan(arctan(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arctan(arctan(x))\right)}{dx}\\=&(\frac{((\frac{(1)}{(1 + (x)^{2})}))}{(1 + (arctan(x))^{2})})\\=&\frac{1}{(x^{2} + 1)(arctan^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{(x^{2} + 1)(arctan^{2}(x) + 1)}\right)}{dx}\\=&\frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{(arctan^{2}(x) + 1)} + \frac{(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})}{(x^{2} + 1)}\\=&\frac{-2x}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2x}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\right)}{dx}\\=&\frac{-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x}{(arctan^{2}(x) + 1)} - \frac{2(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})x}{(x^{2} + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})arctan(x)}{(x^{2} + 1)^{2}} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})arctan(x)}{(arctan^{2}(x) + 1)^{2}} - \frac{2(\frac{(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\\=&\frac{12xarctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8x^{2}}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{12xarctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8x^{2}}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)}\right)}{dx}\\=&\frac{12(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})xarctan(x)}{(x^{2} + 1)^{3}} + \frac{12(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})xarctan(x)}{(arctan^{2}(x) + 1)^{2}} + \frac{12arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{12x(\frac{(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}}{(arctan^{2}(x) + 1)} + \frac{8(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})x^{2}}{(x^{2} + 1)^{3}} + \frac{8*2x}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8(\frac{-3((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{4}})arctan^{2}(x)}{(x^{2} + 1)^{3}} + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}} + \frac{8(\frac{2arctan(x)(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})}{(arctan^{2}(x) + 1)^{2}} - \frac{2(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})}{(x^{2} + 1)^{3}} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})}{(arctan^{2}(x) + 1)} - \frac{2(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})}{(x^{2} + 1)^{2}}\\=& - \frac{96xarctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{4}} - \frac{88x^{2}arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{4}} + \frac{16arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{24x}{(x^{2} + 1)^{4}(arctan^{2}(x) + 1)^{2}} - \frac{48x^{3}}{(x^{2} + 1)^{4}(arctan^{2}(x) + 1)} + \frac{16x}{(arctan^{2}(x) + 1)(x^{2} + 1)^{3}} - \frac{48arctan^{3}(x)}{(arctan^{2}(x) + 1)^{4}(x^{2} + 1)^{4}} + \frac{24arctan(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{4}} + \frac{8x}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)}\\ \end{split}\end{equation} \]
你的问题在这里没有得到解决？请到 热门难题 里面看看吧！
最 新 发 布
新增加身体健康评估计算器，位置：“数学运算 > 身体健康评估”。
新增加学习笔记(安卓版)百度网盘快速下载应用程序，欢迎使用。
新增加学习笔记(安卓版)本站下载应用程序，欢迎使用。
新增线性代数行列式的计算，欢迎使用。
数学计算和一元方程已经支持正割函数和余割函数，欢迎使用。
新增加贷款计算器模块（具体位置：数学运算 > 贷款计算器），欢迎使用。