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 2 个题目：每一题对 x 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/2】求函数\frac{(sin(x) + cos(x))}{(sin(x) - cos(x))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{sin(x)}{(sin(x) - cos(x))} + \frac{cos(x)}{(sin(x) - cos(x))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{sin(x)}{(sin(x) - cos(x))} + \frac{cos(x)}{(sin(x) - cos(x))}\right)}{dx}\\=&(\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})sin(x) + \frac{cos(x)}{(sin(x) - cos(x))} + (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})cos(x) + \frac{-sin(x)}{(sin(x) - cos(x))}\\=&\frac{-2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{cos(x)}{(sin(x) - cos(x))} - \frac{cos^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{sin(x)}{(sin(x) - cos(x))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{cos(x)}{(sin(x) - cos(x))} - \frac{cos^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{sin(x)}{(sin(x) - cos(x))}\right)}{dx}\\=&-2(\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin(x)cos(x) - \frac{2cos(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{2sin(x)*-sin(x)}{(sin(x) - cos(x))^{2}} - (\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin^{2}(x) - \frac{2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} + (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})cos(x) + \frac{-sin(x)}{(sin(x) - cos(x))} - (\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})cos^{2}(x) - \frac{-2cos(x)sin(x)}{(sin(x) - cos(x))^{2}} - (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})sin(x) - \frac{cos(x)}{(sin(x) - cos(x))}\\=&\frac{6sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{6sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{3cos^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{3sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{2sin^{3}(x)}{(sin(x) - cos(x))^{3}} + \frac{2cos^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{sin(x)}{(sin(x) - cos(x))} - \frac{cos(x)}{(sin(x) - cos(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{6sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{6sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{3cos^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{3sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{2sin^{3}(x)}{(sin(x) - cos(x))^{3}} + \frac{2cos^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{sin(x)}{(sin(x) - cos(x))} - \frac{cos(x)}{(sin(x) - cos(x))}\right)}{dx}\\=&6(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin(x)cos^{2}(x) + \frac{6cos(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{6sin(x)*-2cos(x)sin(x)}{(sin(x) - cos(x))^{3}} + 6(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin^{2}(x)cos(x) + \frac{6*2sin(x)cos(x)cos(x)}{(sin(x) - cos(x))^{3}} + \frac{6sin^{2}(x)*-sin(x)}{(sin(x) - cos(x))^{3}} - 3(\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})cos^{2}(x) - \frac{3*-2cos(x)sin(x)}{(sin(x) - cos(x))^{2}} + 3(\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin^{2}(x) + \frac{3*2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} + 2(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin^{3}(x) + \frac{2*3sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} + 2(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})cos^{3}(x) + \frac{2*-3cos^{2}(x)sin(x)}{(sin(x) - cos(x))^{3}} - (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})sin(x) - \frac{cos(x)}{(sin(x) - cos(x))} - (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})cos(x) - \frac{-sin(x)}{(sin(x) - cos(x))}\\=&\frac{-24sin(x)cos^{3}(x)}{(sin(x) - cos(x))^{4}} - \frac{36sin^{2}(x)cos^{2}(x)}{(sin(x) - cos(x))^{4}} + \frac{12cos^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{12sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{24sin^{3}(x)cos(x)}{(sin(x) - cos(x))^{4}} + \frac{12sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{14sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{12sin^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{6sin^{4}(x)}{(sin(x) - cos(x))^{4}} - \frac{6cos^{4}(x)}{(sin(x) - cos(x))^{4}} + \frac{sin^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{cos(x)}{(sin(x) - cos(x))} + \frac{cos^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{sin(x)}{(sin(x) - cos(x))}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-24sin(x)cos^{3}(x)}{(sin(x) - cos(x))^{4}} - \frac{36sin^{2}(x)cos^{2}(x)}{(sin(x) - cos(x))^{4}} + \frac{12cos^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{12sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{24sin^{3}(x)cos(x)}{(sin(x) - cos(x))^{4}} + \frac{12sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{14sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{12sin^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{6sin^{4}(x)}{(sin(x) - cos(x))^{4}} - \frac{6cos^{4}(x)}{(sin(x) - cos(x))^{4}} + \frac{sin^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{cos(x)}{(sin(x) - cos(x))} + \frac{cos^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{sin(x)}{(sin(x) - cos(x))}\right)}{dx}\\=&-24(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})sin(x)cos^{3}(x) - \frac{24cos(x)cos^{3}(x)}{(sin(x) - cos(x))^{4}} - \frac{24sin(x)*-3cos^{2}(x)sin(x)}{(sin(x) - cos(x))^{4}} - 36(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})sin^{2}(x)cos^{2}(x) - \frac{36*2sin(x)cos(x)cos^{2}(x)}{(sin(x) - cos(x))^{4}} - \frac{36sin^{2}(x)*-2cos(x)sin(x)}{(sin(x) - cos(x))^{4}} + 12(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})cos^{3}(x) + \frac{12*-3cos^{2}(x)sin(x)}{(sin(x) - cos(x))^{3}} - 12(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin^{2}(x)cos(x) - \frac{12*2sin(x)cos(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{12sin^{2}(x)*-sin(x)}{(sin(x) - cos(x))^{3}} - 24(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})sin^{3}(x)cos(x) - \frac{24*3sin^{2}(x)cos(x)cos(x)}{(sin(x) - cos(x))^{4}} - \frac{24sin^{3}(x)*-sin(x)}{(sin(x) - cos(x))^{4}} + 12(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin(x)cos^{2}(x) + \frac{12cos(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{12sin(x)*-2cos(x)sin(x)}{(sin(x) - cos(x))^{3}} + 14(\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin(x)cos(x) + \frac{14cos(x)cos(x)}{(sin(x) - cos(x))^{2}} + \frac{14sin(x)*-sin(x)}{(sin(x) - cos(x))^{2}} - 12(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin^{3}(x) - \frac{12*3sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - 6(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})sin^{4}(x) - \frac{6*4sin^{3}(x)cos(x)}{(sin(x) - cos(x))^{4}} - 6(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})cos^{4}(x) - \frac{6*-4cos^{3}(x)sin(x)}{(sin(x) - cos(x))^{4}} + (\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin^{2}(x) + \frac{2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})cos(x) - \frac{-sin(x)}{(sin(x) - cos(x))} + (\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})cos^{2}(x) + \frac{-2cos(x)sin(x)}{(sin(x) - cos(x))^{2}} + (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})sin(x) + \frac{cos(x)}{(sin(x) - cos(x))}\\=&\frac{120sin(x)cos^{4}(x)}{(sin(x) - cos(x))^{5}} + \frac{240sin^{2}(x)cos^{3}(x)}{(sin(x) - cos(x))^{5}} - \frac{60cos^{4}(x)}{(sin(x) - cos(x))^{4}} + \frac{240sin^{3}(x)cos^{2}(x)}{(sin(x) - cos(x))^{5}} - \frac{120sin(x)cos^{3}(x)}{(sin(x) - cos(x))^{4}} + \frac{120sin^{3}(x)cos(x)}{(sin(x) - cos(x))^{4}} - \frac{90sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} - \frac{90sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} + \frac{120sin^{4}(x)cos(x)}{(sin(x) - cos(x))^{5}} + \frac{60sin^{4}(x)}{(sin(x) - cos(x))^{4}} + \frac{10cos^{3}(x)}{(sin(x) - cos(x))^{3}} + \frac{10sin^{3}(x)}{(sin(x) - cos(x))^{3}} + \frac{15cos^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{15sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{24sin^{5}(x)}{(sin(x) - cos(x))^{5}} + \frac{24cos^{5}(x)}{(sin(x) - cos(x))^{5}} + \frac{sin(x)}{(sin(x) - cos(x))} + \frac{cos(x)}{(sin(x) - cos(x))}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数\frac{(ch(x) + sh(x))}{(ch(x) - sh(x))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{ch(x)}{(ch(x) - sh(x))} + \frac{sh(x)}{(ch(x) - sh(x))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{ch(x)}{(ch(x) - sh(x))} + \frac{sh(x)}{(ch(x) - sh(x))}\right)}{dx}\\=&(\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})ch(x) + \frac{sh(x)}{(ch(x) - sh(x))} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})sh(x) + \frac{ch(x)}{(ch(x) - sh(x))}\\=&\frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))}\right)}{dx}\\=&(\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})ch^{2}(x) + \frac{2ch(x)sh(x)}{(ch(x) - sh(x))^{2}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})sh(x) + \frac{ch(x)}{(ch(x) - sh(x))} - (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})sh^{2}(x) - \frac{2sh(x)ch(x)}{(ch(x) - sh(x))^{2}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})ch(x) + \frac{sh(x)}{(ch(x) - sh(x))}\\=& - \frac{2sh(x)ch^{2}(x)}{(ch(x) - sh(x))^{3}} + \frac{2ch^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh^{2}(x)ch(x)}{(ch(x) - sh(x))^{3}} + \frac{ch(x)}{(ch(x) - sh(x))} + \frac{2sh^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{2sh(x)ch^{2}(x)}{(ch(x) - sh(x))^{3}} + \frac{2ch^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh^{2}(x)ch(x)}{(ch(x) - sh(x))^{3}} + \frac{ch(x)}{(ch(x) - sh(x))} + \frac{2sh^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))}\right)}{dx}\\=& - 2(\frac{-3(sh(x) - ch(x))}{(ch(x) - sh(x))^{4}})sh(x)ch^{2}(x) - \frac{2ch(x)ch^{2}(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh(x)*2ch(x)sh(x)}{(ch(x) - sh(x))^{3}} + 2(\frac{-3(sh(x) - ch(x))}{(ch(x) - sh(x))^{4}})ch^{3}(x) + \frac{2*3ch^{2}(x)sh(x)}{(ch(x) - sh(x))^{3}} - 2(\frac{-3(sh(x) - ch(x))}{(ch(x) - sh(x))^{4}})sh^{2}(x)ch(x) - \frac{2*2sh(x)ch(x)ch(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh^{2}(x)sh(x)}{(ch(x) - sh(x))^{3}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})ch(x) + \frac{sh(x)}{(ch(x) - sh(x))} + 2(\frac{-3(sh(x) - ch(x))}{(ch(x) - sh(x))^{4}})sh^{3}(x) + \frac{2*3sh^{2}(x)ch(x)}{(ch(x) - sh(x))^{3}} - (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})sh^{2}(x) - \frac{2sh(x)ch(x)}{(ch(x) - sh(x))^{2}} + (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})ch^{2}(x) + \frac{2ch(x)sh(x)}{(ch(x) - sh(x))^{2}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})sh(x) + \frac{ch(x)}{(ch(x) - sh(x))}\\=& - \frac{12sh(x)ch^{3}(x)}{(ch(x) - sh(x))^{4}} + \frac{6ch^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{12sh^{3}(x)ch(x)}{(ch(x) - sh(x))^{4}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} - \frac{6sh^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{sh(x)}{(ch(x) - sh(x))} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( - \frac{12sh(x)ch^{3}(x)}{(ch(x) - sh(x))^{4}} + \frac{6ch^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{12sh^{3}(x)ch(x)}{(ch(x) - sh(x))^{4}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} - \frac{6sh^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{sh(x)}{(ch(x) - sh(x))} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))}\right)}{dx}\\=& - 12(\frac{-4(sh(x) - ch(x))}{(ch(x) - sh(x))^{5}})sh(x)ch^{3}(x) - \frac{12ch(x)ch^{3}(x)}{(ch(x) - sh(x))^{4}} - \frac{12sh(x)*3ch^{2}(x)sh(x)}{(ch(x) - sh(x))^{4}} + 6(\frac{-4(sh(x) - ch(x))}{(ch(x) - sh(x))^{5}})ch^{4}(x) + \frac{6*4ch^{3}(x)sh(x)}{(ch(x) - sh(x))^{4}} + 12(\frac{-4(sh(x) - ch(x))}{(ch(x) - sh(x))^{5}})sh^{3}(x)ch(x) + \frac{12*3sh^{2}(x)ch(x)ch(x)}{(ch(x) - sh(x))^{4}} + \frac{12sh^{3}(x)sh(x)}{(ch(x) - sh(x))^{4}} + (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})ch^{2}(x) + \frac{2ch(x)sh(x)}{(ch(x) - sh(x))^{2}} - 6(\frac{-4(sh(x) - ch(x))}{(ch(x) - sh(x))^{5}})sh^{4}(x) - \frac{6*4sh^{3}(x)ch(x)}{(ch(x) - sh(x))^{4}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})sh(x) + \frac{ch(x)}{(ch(x) - sh(x))} - (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})sh^{2}(x) - \frac{2sh(x)ch(x)}{(ch(x) - sh(x))^{2}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})ch(x) + \frac{sh(x)}{(ch(x) - sh(x))}\\=&\frac{48sh^{2}(x)ch^{3}(x)}{(ch(x) - sh(x))^{5}} - \frac{72sh(x)ch^{4}(x)}{(ch(x) - sh(x))^{5}} - \frac{12ch^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{48sh^{3}(x)ch^{2}(x)}{(ch(x) - sh(x))^{5}} + \frac{24ch^{5}(x)}{(ch(x) - sh(x))^{5}} - \frac{72sh^{4}(x)ch(x)}{(ch(x) - sh(x))^{5}} + \frac{24sh(x)ch^{3}(x)}{(ch(x) - sh(x))^{4}} - \frac{24sh^{3}(x)ch(x)}{(ch(x) - sh(x))^{4}} - \frac{2sh(x)ch^{2}(x)}{(ch(x) - sh(x))^{3}} + \frac{2ch^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh^{2}(x)ch(x)}{(ch(x) - sh(x))^{3}} + \frac{12sh^{4}(x)}{(ch(x) - sh(x))^{4}} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))} + \frac{2sh^{3}(x)}{(ch(x) - sh(x))^{3}} + \frac{24sh^{5}(x)}{(ch(x) - sh(x))^{5}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/2】求函数\frac{(sin(x) + cos(x))}{(sin(x) - cos(x))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{sin(x)}{(sin(x) - cos(x))} + \frac{cos(x)}{(sin(x) - cos(x))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{sin(x)}{(sin(x) - cos(x))} + \frac{cos(x)}{(sin(x) - cos(x))}\right)}{dx}\\=&(\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})sin(x) + \frac{cos(x)}{(sin(x) - cos(x))} + (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})cos(x) + \frac{-sin(x)}{(sin(x) - cos(x))}\\=&\frac{-2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{cos(x)}{(sin(x) - cos(x))} - \frac{cos^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{sin(x)}{(sin(x) - cos(x))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{cos(x)}{(sin(x) - cos(x))} - \frac{cos^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{sin(x)}{(sin(x) - cos(x))}\right)}{dx}\\=&-2(\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin(x)cos(x) - \frac{2cos(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{2sin(x)*-sin(x)}{(sin(x) - cos(x))^{2}} - (\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin^{2}(x) - \frac{2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} + (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})cos(x) + \frac{-sin(x)}{(sin(x) - cos(x))} - (\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})cos^{2}(x) - \frac{-2cos(x)sin(x)}{(sin(x) - cos(x))^{2}} - (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})sin(x) - \frac{cos(x)}{(sin(x) - cos(x))}\\=&\frac{6sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{6sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{3cos^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{3sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{2sin^{3}(x)}{(sin(x) - cos(x))^{3}} + \frac{2cos^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{sin(x)}{(sin(x) - cos(x))} - \frac{cos(x)}{(sin(x) - cos(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{6sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{6sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{3cos^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{3sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{2sin^{3}(x)}{(sin(x) - cos(x))^{3}} + \frac{2cos^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{sin(x)}{(sin(x) - cos(x))} - \frac{cos(x)}{(sin(x) - cos(x))}\right)}{dx}\\=&6(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin(x)cos^{2}(x) + \frac{6cos(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{6sin(x)*-2cos(x)sin(x)}{(sin(x) - cos(x))^{3}} + 6(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin^{2}(x)cos(x) + \frac{6*2sin(x)cos(x)cos(x)}{(sin(x) - cos(x))^{3}} + \frac{6sin^{2}(x)*-sin(x)}{(sin(x) - cos(x))^{3}} - 3(\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})cos^{2}(x) - \frac{3*-2cos(x)sin(x)}{(sin(x) - cos(x))^{2}} + 3(\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin^{2}(x) + \frac{3*2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} + 2(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin^{3}(x) + \frac{2*3sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} + 2(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})cos^{3}(x) + \frac{2*-3cos^{2}(x)sin(x)}{(sin(x) - cos(x))^{3}} - (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})sin(x) - \frac{cos(x)}{(sin(x) - cos(x))} - (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})cos(x) - \frac{-sin(x)}{(sin(x) - cos(x))}\\=&\frac{-24sin(x)cos^{3}(x)}{(sin(x) - cos(x))^{4}} - \frac{36sin^{2}(x)cos^{2}(x)}{(sin(x) - cos(x))^{4}} + \frac{12cos^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{12sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{24sin^{3}(x)cos(x)}{(sin(x) - cos(x))^{4}} + \frac{12sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{14sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{12sin^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{6sin^{4}(x)}{(sin(x) - cos(x))^{4}} - \frac{6cos^{4}(x)}{(sin(x) - cos(x))^{4}} + \frac{sin^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{cos(x)}{(sin(x) - cos(x))} + \frac{cos^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{sin(x)}{(sin(x) - cos(x))}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-24sin(x)cos^{3}(x)}{(sin(x) - cos(x))^{4}} - \frac{36sin^{2}(x)cos^{2}(x)}{(sin(x) - cos(x))^{4}} + \frac{12cos^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{12sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{24sin^{3}(x)cos(x)}{(sin(x) - cos(x))^{4}} + \frac{12sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{14sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - \frac{12sin^{3}(x)}{(sin(x) - cos(x))^{3}} - \frac{6sin^{4}(x)}{(sin(x) - cos(x))^{4}} - \frac{6cos^{4}(x)}{(sin(x) - cos(x))^{4}} + \frac{sin^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{cos(x)}{(sin(x) - cos(x))} + \frac{cos^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{sin(x)}{(sin(x) - cos(x))}\right)}{dx}\\=&-24(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})sin(x)cos^{3}(x) - \frac{24cos(x)cos^{3}(x)}{(sin(x) - cos(x))^{4}} - \frac{24sin(x)*-3cos^{2}(x)sin(x)}{(sin(x) - cos(x))^{4}} - 36(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})sin^{2}(x)cos^{2}(x) - \frac{36*2sin(x)cos(x)cos^{2}(x)}{(sin(x) - cos(x))^{4}} - \frac{36sin^{2}(x)*-2cos(x)sin(x)}{(sin(x) - cos(x))^{4}} + 12(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})cos^{3}(x) + \frac{12*-3cos^{2}(x)sin(x)}{(sin(x) - cos(x))^{3}} - 12(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin^{2}(x)cos(x) - \frac{12*2sin(x)cos(x)cos(x)}{(sin(x) - cos(x))^{3}} - \frac{12sin^{2}(x)*-sin(x)}{(sin(x) - cos(x))^{3}} - 24(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})sin^{3}(x)cos(x) - \frac{24*3sin^{2}(x)cos(x)cos(x)}{(sin(x) - cos(x))^{4}} - \frac{24sin^{3}(x)*-sin(x)}{(sin(x) - cos(x))^{4}} + 12(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin(x)cos^{2}(x) + \frac{12cos(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} + \frac{12sin(x)*-2cos(x)sin(x)}{(sin(x) - cos(x))^{3}} + 14(\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin(x)cos(x) + \frac{14cos(x)cos(x)}{(sin(x) - cos(x))^{2}} + \frac{14sin(x)*-sin(x)}{(sin(x) - cos(x))^{2}} - 12(\frac{-3(cos(x) - -sin(x))}{(sin(x) - cos(x))^{4}})sin^{3}(x) - \frac{12*3sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} - 6(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})sin^{4}(x) - \frac{6*4sin^{3}(x)cos(x)}{(sin(x) - cos(x))^{4}} - 6(\frac{-4(cos(x) - -sin(x))}{(sin(x) - cos(x))^{5}})cos^{4}(x) - \frac{6*-4cos^{3}(x)sin(x)}{(sin(x) - cos(x))^{4}} + (\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})sin^{2}(x) + \frac{2sin(x)cos(x)}{(sin(x) - cos(x))^{2}} - (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})cos(x) - \frac{-sin(x)}{(sin(x) - cos(x))} + (\frac{-2(cos(x) - -sin(x))}{(sin(x) - cos(x))^{3}})cos^{2}(x) + \frac{-2cos(x)sin(x)}{(sin(x) - cos(x))^{2}} + (\frac{-(cos(x) - -sin(x))}{(sin(x) - cos(x))^{2}})sin(x) + \frac{cos(x)}{(sin(x) - cos(x))}\\=&\frac{120sin(x)cos^{4}(x)}{(sin(x) - cos(x))^{5}} + \frac{240sin^{2}(x)cos^{3}(x)}{(sin(x) - cos(x))^{5}} - \frac{60cos^{4}(x)}{(sin(x) - cos(x))^{4}} + \frac{240sin^{3}(x)cos^{2}(x)}{(sin(x) - cos(x))^{5}} - \frac{120sin(x)cos^{3}(x)}{(sin(x) - cos(x))^{4}} + \frac{120sin^{3}(x)cos(x)}{(sin(x) - cos(x))^{4}} - \frac{90sin(x)cos^{2}(x)}{(sin(x) - cos(x))^{3}} - \frac{90sin^{2}(x)cos(x)}{(sin(x) - cos(x))^{3}} + \frac{120sin^{4}(x)cos(x)}{(sin(x) - cos(x))^{5}} + \frac{60sin^{4}(x)}{(sin(x) - cos(x))^{4}} + \frac{10cos^{3}(x)}{(sin(x) - cos(x))^{3}} + \frac{10sin^{3}(x)}{(sin(x) - cos(x))^{3}} + \frac{15cos^{2}(x)}{(sin(x) - cos(x))^{2}} - \frac{15sin^{2}(x)}{(sin(x) - cos(x))^{2}} + \frac{24sin^{5}(x)}{(sin(x) - cos(x))^{5}} + \frac{24cos^{5}(x)}{(sin(x) - cos(x))^{5}} + \frac{sin(x)}{(sin(x) - cos(x))} + \frac{cos(x)}{(sin(x) - cos(x))}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/2】求函数\frac{(ch(x) + sh(x))}{(ch(x) - sh(x))} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{ch(x)}{(ch(x) - sh(x))} + \frac{sh(x)}{(ch(x) - sh(x))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{ch(x)}{(ch(x) - sh(x))} + \frac{sh(x)}{(ch(x) - sh(x))}\right)}{dx}\\=&(\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})ch(x) + \frac{sh(x)}{(ch(x) - sh(x))} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})sh(x) + \frac{ch(x)}{(ch(x) - sh(x))}\\=&\frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))}\right)}{dx}\\=&(\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})ch^{2}(x) + \frac{2ch(x)sh(x)}{(ch(x) - sh(x))^{2}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})sh(x) + \frac{ch(x)}{(ch(x) - sh(x))} - (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})sh^{2}(x) - \frac{2sh(x)ch(x)}{(ch(x) - sh(x))^{2}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})ch(x) + \frac{sh(x)}{(ch(x) - sh(x))}\\=& - \frac{2sh(x)ch^{2}(x)}{(ch(x) - sh(x))^{3}} + \frac{2ch^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh^{2}(x)ch(x)}{(ch(x) - sh(x))^{3}} + \frac{ch(x)}{(ch(x) - sh(x))} + \frac{2sh^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - \frac{2sh(x)ch^{2}(x)}{(ch(x) - sh(x))^{3}} + \frac{2ch^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh^{2}(x)ch(x)}{(ch(x) - sh(x))^{3}} + \frac{ch(x)}{(ch(x) - sh(x))} + \frac{2sh^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))}\right)}{dx}\\=& - 2(\frac{-3(sh(x) - ch(x))}{(ch(x) - sh(x))^{4}})sh(x)ch^{2}(x) - \frac{2ch(x)ch^{2}(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh(x)*2ch(x)sh(x)}{(ch(x) - sh(x))^{3}} + 2(\frac{-3(sh(x) - ch(x))}{(ch(x) - sh(x))^{4}})ch^{3}(x) + \frac{2*3ch^{2}(x)sh(x)}{(ch(x) - sh(x))^{3}} - 2(\frac{-3(sh(x) - ch(x))}{(ch(x) - sh(x))^{4}})sh^{2}(x)ch(x) - \frac{2*2sh(x)ch(x)ch(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh^{2}(x)sh(x)}{(ch(x) - sh(x))^{3}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})ch(x) + \frac{sh(x)}{(ch(x) - sh(x))} + 2(\frac{-3(sh(x) - ch(x))}{(ch(x) - sh(x))^{4}})sh^{3}(x) + \frac{2*3sh^{2}(x)ch(x)}{(ch(x) - sh(x))^{3}} - (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})sh^{2}(x) - \frac{2sh(x)ch(x)}{(ch(x) - sh(x))^{2}} + (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})ch^{2}(x) + \frac{2ch(x)sh(x)}{(ch(x) - sh(x))^{2}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})sh(x) + \frac{ch(x)}{(ch(x) - sh(x))}\\=& - \frac{12sh(x)ch^{3}(x)}{(ch(x) - sh(x))^{4}} + \frac{6ch^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{12sh^{3}(x)ch(x)}{(ch(x) - sh(x))^{4}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} - \frac{6sh^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{sh(x)}{(ch(x) - sh(x))} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( - \frac{12sh(x)ch^{3}(x)}{(ch(x) - sh(x))^{4}} + \frac{6ch^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{12sh^{3}(x)ch(x)}{(ch(x) - sh(x))^{4}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} - \frac{6sh^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{sh(x)}{(ch(x) - sh(x))} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))}\right)}{dx}\\=& - 12(\frac{-4(sh(x) - ch(x))}{(ch(x) - sh(x))^{5}})sh(x)ch^{3}(x) - \frac{12ch(x)ch^{3}(x)}{(ch(x) - sh(x))^{4}} - \frac{12sh(x)*3ch^{2}(x)sh(x)}{(ch(x) - sh(x))^{4}} + 6(\frac{-4(sh(x) - ch(x))}{(ch(x) - sh(x))^{5}})ch^{4}(x) + \frac{6*4ch^{3}(x)sh(x)}{(ch(x) - sh(x))^{4}} + 12(\frac{-4(sh(x) - ch(x))}{(ch(x) - sh(x))^{5}})sh^{3}(x)ch(x) + \frac{12*3sh^{2}(x)ch(x)ch(x)}{(ch(x) - sh(x))^{4}} + \frac{12sh^{3}(x)sh(x)}{(ch(x) - sh(x))^{4}} + (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})ch^{2}(x) + \frac{2ch(x)sh(x)}{(ch(x) - sh(x))^{2}} - 6(\frac{-4(sh(x) - ch(x))}{(ch(x) - sh(x))^{5}})sh^{4}(x) - \frac{6*4sh^{3}(x)ch(x)}{(ch(x) - sh(x))^{4}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})sh(x) + \frac{ch(x)}{(ch(x) - sh(x))} - (\frac{-2(sh(x) - ch(x))}{(ch(x) - sh(x))^{3}})sh^{2}(x) - \frac{2sh(x)ch(x)}{(ch(x) - sh(x))^{2}} + (\frac{-(sh(x) - ch(x))}{(ch(x) - sh(x))^{2}})ch(x) + \frac{sh(x)}{(ch(x) - sh(x))}\\=&\frac{48sh^{2}(x)ch^{3}(x)}{(ch(x) - sh(x))^{5}} - \frac{72sh(x)ch^{4}(x)}{(ch(x) - sh(x))^{5}} - \frac{12ch^{4}(x)}{(ch(x) - sh(x))^{4}} + \frac{48sh^{3}(x)ch^{2}(x)}{(ch(x) - sh(x))^{5}} + \frac{24ch^{5}(x)}{(ch(x) - sh(x))^{5}} - \frac{72sh^{4}(x)ch(x)}{(ch(x) - sh(x))^{5}} + \frac{24sh(x)ch^{3}(x)}{(ch(x) - sh(x))^{4}} - \frac{24sh^{3}(x)ch(x)}{(ch(x) - sh(x))^{4}} - \frac{2sh(x)ch^{2}(x)}{(ch(x) - sh(x))^{3}} + \frac{2ch^{3}(x)}{(ch(x) - sh(x))^{3}} - \frac{2sh^{2}(x)ch(x)}{(ch(x) - sh(x))^{3}} + \frac{12sh^{4}(x)}{(ch(x) - sh(x))^{4}} - \frac{sh^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{ch(x)}{(ch(x) - sh(x))} + \frac{2sh^{3}(x)}{(ch(x) - sh(x))^{3}} + \frac{24sh^{5}(x)}{(ch(x) - sh(x))^{5}} + \frac{ch^{2}(x)}{(ch(x) - sh(x))^{2}} + \frac{sh(x)}{(ch(x) - sh(x))}\\ \end{split}\end{equation} \]
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