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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(tan(x) + tan(y))}{(1 - tan(x)tan(y))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{tan(x)}{(-tan(x)tan(y) + 1)} + \frac{tan(y)}{(-tan(x)tan(y) + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{tan(x)}{(-tan(x)tan(y) + 1)} + \frac{tan(y)}{(-tan(x)tan(y) + 1)}\right)}{dx}\\=&(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(x) + \frac{sec^{2}(x)(1)}{(-tan(x)tan(y) + 1)} + (\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(y) + \frac{sec^{2}(y)(0)}{(-tan(x)tan(y) + 1)}\\=&\frac{tan(y)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{tan(y)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan(x)sec^{2}(x) + \frac{sec^{2}(y)(0)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan(y)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan(y)tan(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + (\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})sec^{2}(x) + \frac{2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + (\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(y)sec^{2}(x) + \frac{2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{2tan^{2}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2tan^{2}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&2(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)tan(x)sec^{4}(x) + \frac{2*2tan(y)sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)sec^{4}(x) + \frac{2sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan^{2}(x)sec^{2}(x) + \frac{2sec^{2}(y)(0)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{3}(y)sec^{4}(x) + \frac{2*3tan^{2}(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{3}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(x)sec^{2}(x) + \frac{2sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(x)tan^{2}(y)sec^{2}(x) + \frac{2sec^{2}(x)(1)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(x)*2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(x)tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{6tan^{3}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{6tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{8tan(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{3}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{6tan^{3}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{6tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{8tan(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{3}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&6(\frac{-4(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{5}})tan^{3}(y)tan(x)sec^{6}(x) + \frac{6*3tan^{2}(y)sec^{2}(y)(0)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{3}(y)sec^{2}(x)(1)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{3}(y)tan(x)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{4}} + 6(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)sec^{6}(x) + \frac{6*2tan(y)sec^{2}(y)(0)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{6tan^{2}(y)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 12(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)tan^{2}(x)sec^{4}(x) + \frac{12*2tan(y)sec^{2}(y)(0)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)*2tan(x)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 12(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(x)tan(y)sec^{4}(x) + \frac{12sec^{2}(x)(1)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{12tan(x)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{12tan(x)tan(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 4(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan^{3}(x)sec^{2}(x) + \frac{4sec^{2}(y)(0)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 6(\frac{-4(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{5}})tan^{4}(y)sec^{6}(x) + \frac{6*4tan^{3}(y)sec^{2}(y)(0)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{4}(y)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{4}} + 8(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan(x)tan^{3}(y)sec^{4}(x) + \frac{8sec^{2}(x)(1)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan(x)*3tan^{2}(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan(x)tan^{3}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan(x)sec^{4}(x) + \frac{2sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})sec^{4}(x) + \frac{2*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 4(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan^{2}(x)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 4(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{3}(y)tan(x)sec^{4}(x) + \frac{4*3tan^{2}(y)sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{4tan^{3}(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{4tan^{3}(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(y)sec^{4}(x) + \frac{2*2tan(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{2}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 4(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(x)tan^{2}(y)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)*2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{24tan^{4}(y)tan(x)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{5}} + \frac{24tan^{3}(y)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{72tan^{3}(y)tan^{2}(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{60tan(x)tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{60tan^{2}(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{56tan^{2}(y)tan^{3}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{28tan^{2}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{16tan(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{8tan(y)tan^{4}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{24tan^{5}(y)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{5}} + \frac{36tan^{4}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{16tan^{3}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{16tan(x)tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{32tan^{2}(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{16tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{36tan(x)tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{24tan^{3}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{8tan^{3}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(tan(x) + tan(y))}{(1 - tan(x)tan(y))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{tan(x)}{(-tan(x)tan(y) + 1)} + \frac{tan(y)}{(-tan(x)tan(y) + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{tan(x)}{(-tan(x)tan(y) + 1)} + \frac{tan(y)}{(-tan(x)tan(y) + 1)}\right)}{dx}\\=&(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(x) + \frac{sec^{2}(x)(1)}{(-tan(x)tan(y) + 1)} + (\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(y) + \frac{sec^{2}(y)(0)}{(-tan(x)tan(y) + 1)}\\=&\frac{tan(y)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{tan(y)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan(x)sec^{2}(x) + \frac{sec^{2}(y)(0)tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan(y)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan(y)tan(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + (\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})sec^{2}(x) + \frac{2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + (\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(y)sec^{2}(x) + \frac{2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{2tan^{2}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{2tan^{2}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&2(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)tan(x)sec^{4}(x) + \frac{2*2tan(y)sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)sec^{4}(x) + \frac{2sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan^{2}(x)sec^{2}(x) + \frac{2sec^{2}(y)(0)tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan^{2}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{3}(y)sec^{4}(x) + \frac{2*3tan^{2}(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{3}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan(x)sec^{2}(x) + \frac{2sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{2tan(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(x)tan^{2}(y)sec^{2}(x) + \frac{2sec^{2}(x)(1)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(x)*2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(x)tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{6tan^{3}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{6tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{8tan(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{3}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{6tan^{3}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{6tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{8tan(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{3}(y)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{2tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\right)}{dx}\\=&6(\frac{-4(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{5}})tan^{3}(y)tan(x)sec^{6}(x) + \frac{6*3tan^{2}(y)sec^{2}(y)(0)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{3}(y)sec^{2}(x)(1)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{3}(y)tan(x)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{4}} + 6(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)sec^{6}(x) + \frac{6*2tan(y)sec^{2}(y)(0)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{6tan^{2}(y)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 12(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{2}(y)tan^{2}(x)sec^{4}(x) + \frac{12*2tan(y)sec^{2}(y)(0)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)*2tan(x)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan^{2}(y)tan^{2}(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 12(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(x)tan(y)sec^{4}(x) + \frac{12sec^{2}(x)(1)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{12tan(x)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{12tan(x)tan(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 4(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan^{3}(x)sec^{2}(x) + \frac{4sec^{2}(y)(0)tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)*3tan^{2}(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan(y)tan^{3}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 6(\frac{-4(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{5}})tan^{4}(y)sec^{6}(x) + \frac{6*4tan^{3}(y)sec^{2}(y)(0)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{6tan^{4}(y)*6sec^{6}(x)tan(x)}{(-tan(x)tan(y) + 1)^{4}} + 8(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan(x)tan^{3}(y)sec^{4}(x) + \frac{8sec^{2}(x)(1)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan(x)*3tan^{2}(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan(x)tan^{3}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan(y)tan(x)sec^{4}(x) + \frac{2sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 2(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})sec^{4}(x) + \frac{2*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 4(\frac{-(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{2}})tan^{2}(x)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{4tan^{2}(x)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)} + 4(\frac{-3(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{4}})tan^{3}(y)tan(x)sec^{4}(x) + \frac{4*3tan^{2}(y)sec^{2}(y)(0)tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{4tan^{3}(y)sec^{2}(x)(1)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{4tan^{3}(y)tan(x)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{3}} + 2(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(y)sec^{4}(x) + \frac{2*2tan(y)sec^{2}(y)(0)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{2tan^{2}(y)*4sec^{4}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}} + 4(\frac{-2(-sec^{2}(x)(1)tan(y) - tan(x)sec^{2}(y)(0) + 0)}{(-tan(x)tan(y) + 1)^{3}})tan^{2}(x)tan^{2}(y)sec^{2}(x) + \frac{4*2tan(x)sec^{2}(x)(1)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)*2tan(y)sec^{2}(y)(0)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{4tan^{2}(x)tan^{2}(y)*2sec^{2}(x)tan(x)}{(-tan(x)tan(y) + 1)^{2}}\\=&\frac{24tan^{4}(y)tan(x)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{5}} + \frac{24tan^{3}(y)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{72tan^{3}(y)tan^{2}(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{60tan(x)tan^{2}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{60tan^{2}(x)tan(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{56tan^{2}(y)tan^{3}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{28tan^{2}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{16tan(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{8tan(y)tan^{4}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{24tan^{5}(y)sec^{8}(x)}{(-tan(x)tan(y) + 1)^{5}} + \frac{36tan^{4}(y)tan(x)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{16tan^{3}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{16tan(x)tan^{2}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{32tan^{2}(x)tan^{3}(y)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{12tan(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{2}} + \frac{16tan(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)} + \frac{36tan(x)tan^{4}(y)sec^{6}(x)}{(-tan(x)tan(y) + 1)^{4}} + \frac{24tan^{3}(y)tan^{2}(x)sec^{4}(x)}{(-tan(x)tan(y) + 1)^{3}} + \frac{8tan^{3}(x)sec^{2}(x)}{(-tan(x)tan(y) + 1)} + \frac{8tan^{3}(x)tan^{2}(y)sec^{2}(x)}{(-tan(x)tan(y) + 1)^{2}}\\ \end{split}\end{equation} \]
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