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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{(sin(X) - sinh(X))}{(cos(X) + cosh(X))} 关于 X 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{sin(X)}{(cos(X) + cosh(X))} - \frac{sinh(X)}{(cos(X) + cosh(X))}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{sin(X)}{(cos(X) + cosh(X))} - \frac{sinh(X)}{(cos(X) + cosh(X))}\right)}{dX}\\=&(\frac{-(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{2}})sin(X) + \frac{cos(X)}{(cos(X) + cosh(X))} - (\frac{-(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{2}})sinh(X) - \frac{cosh(X)}{(cos(X) + cosh(X))}\\=& - \frac{2sin(X)sinh(X)}{(cos(X) + cosh(X))^{2}} + \frac{sin^{2}(X)}{(cos(X) + cosh(X))^{2}} + \frac{cos(X)}{(cos(X) + cosh(X))} + \frac{sinh^{2}(X)}{(cos(X) + cosh(X))^{2}} - \frac{cosh(X)}{(cos(X) + cosh(X))}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( - \frac{2sin(X)sinh(X)}{(cos(X) + cosh(X))^{2}} + \frac{sin^{2}(X)}{(cos(X) + cosh(X))^{2}} + \frac{cos(X)}{(cos(X) + cosh(X))} + \frac{sinh^{2}(X)}{(cos(X) + cosh(X))^{2}} - \frac{cosh(X)}{(cos(X) + cosh(X))}\right)}{dX}\\=& - 2(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})sin(X)sinh(X) - \frac{2cos(X)sinh(X)}{(cos(X) + cosh(X))^{2}} - \frac{2sin(X)cosh(X)}{(cos(X) + cosh(X))^{2}} + (\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})sin^{2}(X) + \frac{2sin(X)cos(X)}{(cos(X) + cosh(X))^{2}} + (\frac{-(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{2}})cos(X) + \frac{-sin(X)}{(cos(X) + cosh(X))} + (\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})sinh^{2}(X) + \frac{2sinh(X)cosh(X)}{(cos(X) + cosh(X))^{2}} - (\frac{-(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{2}})cosh(X) - \frac{sinh(X)}{(cos(X) + cosh(X))}\\=&\frac{-6sin^{2}(X)sinh(X)}{(cos(X) + cosh(X))^{3}} + \frac{6sin(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{3}} - \frac{3cos(X)sinh(X)}{(cos(X) + cosh(X))^{2}} - \frac{3sin(X)cosh(X)}{(cos(X) + cosh(X))^{2}} + \frac{3sin(X)cos(X)}{(cos(X) + cosh(X))^{2}} + \frac{2sin^{3}(X)}{(cos(X) + cosh(X))^{3}} - \frac{sin(X)}{(cos(X) + cosh(X))} + \frac{3sinh(X)cosh(X)}{(cos(X) + cosh(X))^{2}} - \frac{2sinh^{3}(X)}{(cos(X) + cosh(X))^{3}} - \frac{sinh(X)}{(cos(X) + cosh(X))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-6sin^{2}(X)sinh(X)}{(cos(X) + cosh(X))^{3}} + \frac{6sin(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{3}} - \frac{3cos(X)sinh(X)}{(cos(X) + cosh(X))^{2}} - \frac{3sin(X)cosh(X)}{(cos(X) + cosh(X))^{2}} + \frac{3sin(X)cos(X)}{(cos(X) + cosh(X))^{2}} + \frac{2sin^{3}(X)}{(cos(X) + cosh(X))^{3}} - \frac{sin(X)}{(cos(X) + cosh(X))} + \frac{3sinh(X)cosh(X)}{(cos(X) + cosh(X))^{2}} - \frac{2sinh^{3}(X)}{(cos(X) + cosh(X))^{3}} - \frac{sinh(X)}{(cos(X) + cosh(X))}\right)}{dX}\\=&-6(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sin^{2}(X)sinh(X) - \frac{6*2sin(X)cos(X)sinh(X)}{(cos(X) + cosh(X))^{3}} - \frac{6sin^{2}(X)cosh(X)}{(cos(X) + cosh(X))^{3}} + 6(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sin(X)sinh^{2}(X) + \frac{6cos(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{3}} + \frac{6sin(X)*2sinh(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - 3(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})cos(X)sinh(X) - \frac{3*-sin(X)sinh(X)}{(cos(X) + cosh(X))^{2}} - \frac{3cos(X)cosh(X)}{(cos(X) + cosh(X))^{2}} - 3(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})sin(X)cosh(X) - \frac{3cos(X)cosh(X)}{(cos(X) + cosh(X))^{2}} - \frac{3sin(X)sinh(X)}{(cos(X) + cosh(X))^{2}} + 3(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})sin(X)cos(X) + \frac{3cos(X)cos(X)}{(cos(X) + cosh(X))^{2}} + \frac{3sin(X)*-sin(X)}{(cos(X) + cosh(X))^{2}} + 2(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sin^{3}(X) + \frac{2*3sin^{2}(X)cos(X)}{(cos(X) + cosh(X))^{3}} - (\frac{-(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{2}})sin(X) - \frac{cos(X)}{(cos(X) + cosh(X))} + 3(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})sinh(X)cosh(X) + \frac{3cosh(X)cosh(X)}{(cos(X) + cosh(X))^{2}} + \frac{3sinh(X)sinh(X)}{(cos(X) + cosh(X))^{2}} - 2(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sinh^{3}(X) - \frac{2*3sinh^{2}(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - (\frac{-(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{2}})sinh(X) - \frac{cosh(X)}{(cos(X) + cosh(X))}\\=&\frac{24sin(X)sinh(X)cosh(X)}{(cos(X) + cosh(X))^{3}} + \frac{36sin^{2}(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{4}} - \frac{24sin(X)cos(X)sinh(X)}{(cos(X) + cosh(X))^{3}} - \frac{12sin^{2}(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - \frac{24sin(X)sinh^{3}(X)}{(cos(X) + cosh(X))^{4}} + \frac{12cos(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{3}} - \frac{24sin^{3}(X)sinh(X)}{(cos(X) + cosh(X))^{4}} - \frac{6cos(X)cosh(X)}{(cos(X) + cosh(X))^{2}} + \frac{12sin^{2}(X)cos(X)}{(cos(X) + cosh(X))^{3}} + \frac{3cos^{2}(X)}{(cos(X) + cosh(X))^{2}} - \frac{4sin^{2}(X)}{(cos(X) + cosh(X))^{2}} + \frac{6sin^{4}(X)}{(cos(X) + cosh(X))^{4}} - \frac{cos(X)}{(cos(X) + cosh(X))} - \frac{12sinh^{2}(X)cosh(X)}{(cos(X) + cosh(X))^{3}} + \frac{3cosh^{2}(X)}{(cos(X) + cosh(X))^{2}} + \frac{4sinh^{2}(X)}{(cos(X) + cosh(X))^{2}} + \frac{6sinh^{4}(X)}{(cos(X) + cosh(X))^{4}} - \frac{cosh(X)}{(cos(X) + cosh(X))}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{24sin(X)sinh(X)cosh(X)}{(cos(X) + cosh(X))^{3}} + \frac{36sin^{2}(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{4}} - \frac{24sin(X)cos(X)sinh(X)}{(cos(X) + cosh(X))^{3}} - \frac{12sin^{2}(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - \frac{24sin(X)sinh^{3}(X)}{(cos(X) + cosh(X))^{4}} + \frac{12cos(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{3}} - \frac{24sin^{3}(X)sinh(X)}{(cos(X) + cosh(X))^{4}} - \frac{6cos(X)cosh(X)}{(cos(X) + cosh(X))^{2}} + \frac{12sin^{2}(X)cos(X)}{(cos(X) + cosh(X))^{3}} + \frac{3cos^{2}(X)}{(cos(X) + cosh(X))^{2}} - \frac{4sin^{2}(X)}{(cos(X) + cosh(X))^{2}} + \frac{6sin^{4}(X)}{(cos(X) + cosh(X))^{4}} - \frac{cos(X)}{(cos(X) + cosh(X))} - \frac{12sinh^{2}(X)cosh(X)}{(cos(X) + cosh(X))^{3}} + \frac{3cosh^{2}(X)}{(cos(X) + cosh(X))^{2}} + \frac{4sinh^{2}(X)}{(cos(X) + cosh(X))^{2}} + \frac{6sinh^{4}(X)}{(cos(X) + cosh(X))^{4}} - \frac{cosh(X)}{(cos(X) + cosh(X))}\right)}{dX}\\=&24(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sin(X)sinh(X)cosh(X) + \frac{24cos(X)sinh(X)cosh(X)}{(cos(X) + cosh(X))^{3}} + \frac{24sin(X)cosh(X)cosh(X)}{(cos(X) + cosh(X))^{3}} + \frac{24sin(X)sinh(X)sinh(X)}{(cos(X) + cosh(X))^{3}} + 36(\frac{-4(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{5}})sin^{2}(X)sinh^{2}(X) + \frac{36*2sin(X)cos(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{4}} + \frac{36sin^{2}(X)*2sinh(X)cosh(X)}{(cos(X) + cosh(X))^{4}} - 24(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sin(X)cos(X)sinh(X) - \frac{24cos(X)cos(X)sinh(X)}{(cos(X) + cosh(X))^{3}} - \frac{24sin(X)*-sin(X)sinh(X)}{(cos(X) + cosh(X))^{3}} - \frac{24sin(X)cos(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - 12(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sin^{2}(X)cosh(X) - \frac{12*2sin(X)cos(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - \frac{12sin^{2}(X)sinh(X)}{(cos(X) + cosh(X))^{3}} - 24(\frac{-4(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{5}})sin(X)sinh^{3}(X) - \frac{24cos(X)sinh^{3}(X)}{(cos(X) + cosh(X))^{4}} - \frac{24sin(X)*3sinh^{2}(X)cosh(X)}{(cos(X) + cosh(X))^{4}} + 12(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})cos(X)sinh^{2}(X) + \frac{12*-sin(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{3}} + \frac{12cos(X)*2sinh(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - 24(\frac{-4(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{5}})sin^{3}(X)sinh(X) - \frac{24*3sin^{2}(X)cos(X)sinh(X)}{(cos(X) + cosh(X))^{4}} - \frac{24sin^{3}(X)cosh(X)}{(cos(X) + cosh(X))^{4}} - 6(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})cos(X)cosh(X) - \frac{6*-sin(X)cosh(X)}{(cos(X) + cosh(X))^{2}} - \frac{6cos(X)sinh(X)}{(cos(X) + cosh(X))^{2}} + 12(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sin^{2}(X)cos(X) + \frac{12*2sin(X)cos(X)cos(X)}{(cos(X) + cosh(X))^{3}} + \frac{12sin^{2}(X)*-sin(X)}{(cos(X) + cosh(X))^{3}} + 3(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})cos^{2}(X) + \frac{3*-2cos(X)sin(X)}{(cos(X) + cosh(X))^{2}} - 4(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})sin^{2}(X) - \frac{4*2sin(X)cos(X)}{(cos(X) + cosh(X))^{2}} + 6(\frac{-4(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{5}})sin^{4}(X) + \frac{6*4sin^{3}(X)cos(X)}{(cos(X) + cosh(X))^{4}} - (\frac{-(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{2}})cos(X) - \frac{-sin(X)}{(cos(X) + cosh(X))} - 12(\frac{-3(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{4}})sinh^{2}(X)cosh(X) - \frac{12*2sinh(X)cosh(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - \frac{12sinh^{2}(X)sinh(X)}{(cos(X) + cosh(X))^{3}} + 3(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})cosh^{2}(X) + \frac{3*2cosh(X)sinh(X)}{(cos(X) + cosh(X))^{2}} + 4(\frac{-2(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{3}})sinh^{2}(X) + \frac{4*2sinh(X)cosh(X)}{(cos(X) + cosh(X))^{2}} + 6(\frac{-4(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{5}})sinh^{4}(X) + \frac{6*4sinh^{3}(X)cosh(X)}{(cos(X) + cosh(X))^{4}} - (\frac{-(-sin(X) + sinh(X))}{(cos(X) + cosh(X))^{2}})cosh(X) - \frac{sinh(X)}{(cos(X) + cosh(X))}\\=&\frac{180sin^{2}(X)sinh(X)cosh(X)}{(cos(X) + cosh(X))^{4}} - \frac{180sin(X)sinh^{2}(X)cosh(X)}{(cos(X) + cosh(X))^{4}} + \frac{60cos(X)sinh(X)cosh(X)}{(cos(X) + cosh(X))^{3}} + \frac{30sin(X)cosh^{2}(X)}{(cos(X) + cosh(X))^{3}} + \frac{20sin(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{3}} + \frac{240sin^{3}(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{5}} - \frac{240sin^{2}(X)sinh^{3}(X)}{(cos(X) + cosh(X))^{5}} + \frac{180sin(X)cos(X)sinh^{2}(X)}{(cos(X) + cosh(X))^{4}} - \frac{180sin^{2}(X)cos(X)sinh(X)}{(cos(X) + cosh(X))^{4}} - \frac{30cos^{2}(X)sinh(X)}{(cos(X) + cosh(X))^{3}} + \frac{20sin^{2}(X)sinh(X)}{(cos(X) + cosh(X))^{3}} - \frac{60sin(X)cos(X)cosh(X)}{(cos(X) + cosh(X))^{3}} - \frac{60sin^{3}(X)cosh(X)}{(cos(X) + cosh(X))^{4}} + \frac{120sin(X)sinh^{4}(X)}{(cos(X) + cosh(X))^{5}} - \frac{60cos(X)sinh^{3}(X)}{(cos(X) + cosh(X))^{4}} - \frac{120sin^{4}(X)sinh(X)}{(cos(X) + cosh(X))^{5}} + \frac{5sin(X)cosh(X)}{(cos(X) + cosh(X))^{2}} - \frac{5cos(X)sinh(X)}{(cos(X) + cosh(X))^{2}} + \frac{60sin^{3}(X)cos(X)}{(cos(X) + cosh(X))^{4}} + \frac{30sin(X)cos^{2}(X)}{(cos(X) + cosh(X))^{3}} - \frac{15sin(X)cos(X)}{(cos(X) + cosh(X))^{2}} - \frac{20sin^{3}(X)}{(cos(X) + cosh(X))^{3}} + \frac{24sin^{5}(X)}{(cos(X) + cosh(X))^{5}} + \frac{sin(X)}{(cos(X) + cosh(X))} + \frac{60sinh^{3}(X)cosh(X)}{(cos(X) + cosh(X))^{4}} - \frac{30sinh(X)cosh^{2}(X)}{(cos(X) + cosh(X))^{3}} + \frac{15sinh(X)cosh(X)}{(cos(X) + cosh(X))^{2}} - \frac{20sinh^{3}(X)}{(cos(X) + cosh(X))^{3}} - \frac{24sinh^{5}(X)}{(cos(X) + cosh(X))^{5}} - \frac{sinh(X)}{(cos(X) + cosh(X))}\\ \end{split}\end{equation} \]

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