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 4 个题目：每一题对 x 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/4】求函数arcsin(sinh(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arcsin(sinh(x))\right)}{dx}\\=&(\frac{(cosh(x))}{((1 - (sinh(x))^{2})^{\frac{1}{2}})})\\=&\frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}})cosh(x) + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{sinh(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{sinh(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}})sinh(x)cosh^{2}(x) + \frac{cosh(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sinh(x)*2cosh(x)sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}})sinh(x) + \frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{3sinh^{2}(x)cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sinh^{2}(x)cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3sinh^{2}(x)cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sinh^{2}(x)cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{7}{2}}})sinh^{2}(x)cosh^{3}(x) + \frac{3*2sinh(x)cosh(x)cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{3sinh^{2}(x)*3cosh^{2}(x)sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}})cosh^{3}(x) + \frac{3cosh^{2}(x)sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + 3(\frac{\frac{-3}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}})sinh^{2}(x)cosh(x) + \frac{3*2sinh(x)cosh(x)cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sinh^{2}(x)sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}})cosh(x) + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{15sinh^{3}(x)cosh^{4}(x)}{(-sinh^{2}(x) + 1)^{\frac{7}{2}}} + \frac{9sinh(x)cosh^{4}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{18sinh^{3}(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{10sinh(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sinh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/4】求函数arccos(cosh(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arccos(cosh(x))\right)}{dx}\\=&(\frac{-(sinh(x))}{((1 - (cosh(x))^{2})^{\frac{1}{2}})})\\=&\frac{-sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}})sinh(x) - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-sinh^{2}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-sinh^{2}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}})sinh^{2}(x)cosh(x) - \frac{2sinh(x)cosh(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh^{2}(x)sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}})cosh(x) - \frac{sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-3sinh^{3}(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sinh(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-3sinh^{3}(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sinh(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{7}{2}}})sinh^{3}(x)cosh^{2}(x) - \frac{3*3sinh^{2}(x)cosh(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sinh^{3}(x)*2cosh(x)sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}})sinh(x)cosh^{2}(x) - \frac{3cosh(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sinh(x)*2cosh(x)sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-3}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}})sinh^{3}(x) - \frac{3sinh^{2}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}})sinh(x) - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-15sinh^{4}(x)cosh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{7}{2}}} - \frac{18sinh^{2}(x)cosh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{9sinh^{4}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3cosh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{10sinh^{2}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【3/4】求函数arctan(tanh(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arctan(tanh(x))\right)}{dx}\\=&(\frac{(sech^{2}(x))}{(1 + (tanh(x))^{2})})\\=&\frac{sech^{2}(x)}{(tanh^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{sech^{2}(x)}{(tanh^{2}(x) + 1)}\right)}{dx}\\=&(\frac{-(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{2}})sech^{2}(x) + \frac{-2sech(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)}\\=&\frac{-2tanh(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2tanh(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2tanh(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2tanh(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\right)}{dx}\\=&-2(\frac{-2(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{3}})tanh(x)sech^{4}(x) - \frac{2sech^{2}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2tanh(x)*-4sech^{3}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)^{2}} - 2(\frac{-(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{2}})tanh(x)sech^{2}(x) - \frac{2sech^{2}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)} - \frac{2tanh(x)*-2sech(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)}\\=&\frac{8tanh^{2}(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{3}} - \frac{2sech^{6}(x)}{(tanh^{2}(x) + 1)^{2}} + \frac{12tanh^{2}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2sech^{4}(x)}{(tanh^{2}(x) + 1)} + \frac{4tanh^{2}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{8tanh^{2}(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{3}} - \frac{2sech^{6}(x)}{(tanh^{2}(x) + 1)^{2}} + \frac{12tanh^{2}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2sech^{4}(x)}{(tanh^{2}(x) + 1)} + \frac{4tanh^{2}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\right)}{dx}\\=&8(\frac{-3(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{4}})tanh^{2}(x)sech^{6}(x) + \frac{8*2tanh(x)sech^{2}(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{3}} + \frac{8tanh^{2}(x)*-6sech^{5}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)^{3}} - 2(\frac{-2(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{3}})sech^{6}(x) - \frac{2*-6sech^{5}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)^{2}} + 12(\frac{-2(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{3}})tanh^{2}(x)sech^{4}(x) + \frac{12*2tanh(x)sech^{2}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} + \frac{12tanh^{2}(x)*-4sech^{3}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)^{2}} - 2(\frac{-(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{2}})sech^{4}(x) - \frac{2*-4sech^{3}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)} + 4(\frac{-(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{2}})tanh^{2}(x)sech^{2}(x) + \frac{4*2tanh(x)sech^{2}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)} + \frac{4tanh^{2}(x)*-2sech(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)}\\=&\frac{-48tanh^{3}(x)sech^{8}(x)}{(tanh^{2}(x) + 1)^{4}} + \frac{24tanh(x)sech^{8}(x)}{(tanh^{2}(x) + 1)^{3}} - \frac{96tanh^{3}(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{3}} + \frac{40tanh(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{56tanh^{3}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} + \frac{16tanh(x)sech^{4}(x)}{(tanh^{2}(x) + 1)} - \frac{8tanh^{3}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【4/4】求函数arccot(coth(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arccot(coth(x))\right)}{dx}\\=&(\frac{(-csch^{2}(x))tanh(x)sech^{8}(x)}{(1 + (coth(x))^{2})})\\=&\frac{-tanh(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-tanh(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\right)}{dx}\\=&\frac{-(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)sech^{8}(x)csch^{2}(x)}{} + \frac{-sech^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-tanh(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-tanh(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)}\\=&\frac{-2tanh(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2tanh(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\right)}{dx}\\=&\frac{-2(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh(x)coth(x)sech^{8}(x)csch^{4}(x)}{} + \frac{-2sech^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)*-csch^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)coth(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)coth(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})sech^{10}(x)csch^{2}(x)}{} + \frac{--10sech^{9}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-sech^{10}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{8(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh^{2}(x)sech^{8}(x)csch^{2}(x)}{} + \frac{8*2tanh(x)sech^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh^{2}(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{2(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)coth(x)sech^{8}(x)csch^{2}(x)}{} + \frac{2sech^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)*-csch^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)coth(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)coth(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)}\\=&\frac{-8tanh(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-4coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2tanh(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{10tanh(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh^{3}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-8tanh(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-4coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2tanh(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{10tanh(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh^{3}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\right)}{dx}\\=&\frac{-8(\frac{-3(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{4}})tanh(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{} + \frac{-8sech^{2}(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-8tanh(x)*-2coth(x)csch^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-8tanh(x)coth^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-8tanh(x)coth^{2}(x)sech^{8}(x)*-6csch^{5}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-4(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})coth(x)sech^{10}(x)csch^{4}(x)}{} + \frac{-4*-csch^{2}(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-4coth(x)*-10sech^{9}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-4coth(x)sech^{10}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh(x)sech^{8}(x)csch^{6}(x)}{} + \frac{2sech^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2tanh(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2tanh(x)sech^{8}(x)*-6csch^{5}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{} + \frac{32*2tanh(x)sech^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)*-csch^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)coth(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)coth(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{} + \frac{8sech^{2}(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)*-2coth(x)csch^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth^{2}(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{10(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)sech^{10}(x)csch^{2}(x)}{} + \frac{10sech^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{10tanh(x)*-10sech^{9}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{10tanh(x)sech^{10}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{4(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})coth(x)sech^{10}(x)csch^{2}(x)}{} + \frac{4*-csch^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4coth(x)*-10sech^{9}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4coth(x)sech^{10}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{16(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)sech^{10}(x)csch^{2}(x)}{} + \frac{16sech^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)*-10sech^{9}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)sech^{10}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{-64(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh^{3}(x)sech^{8}(x)csch^{2}(x)}{} + \frac{-64*3tanh^{2}(x)sech^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh^{3}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh^{3}(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{-32(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{} + \frac{-32*2tanh(x)sech^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)*-csch^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)coth(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)coth(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{4(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{} + \frac{4sech^{2}(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{4tanh(x)*-2coth(x)csch^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{4tanh(x)coth^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{4tanh(x)coth^{2}(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)sech^{8}(x)csch^{4}(x)}{} + \frac{-2sech^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-2tanh(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-2tanh(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{-4(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{} + \frac{-4sech^{2}(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)*-2coth(x)csch^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)coth^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)coth^{2}(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)}\\=&\frac{-48tanh(x)coth^{3}(x)sech^{8}(x)csch^{8}(x)}{(coth^{2}(x) + 1)^{4}} + \frac{-8coth^{2}(x)sech^{10}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{16tanh(x)coth(x)sech^{8}(x)csch^{8}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{64tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{48tanh(x)coth^{3}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-16coth^{2}(x)sech^{10}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{4sech^{10}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{40tanh(x)coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32coth^{2}(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth(x)sech^{8}(x)csch^{8}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{2sech^{10}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-16tanh^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-12tanh(x)coth(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{128tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{64tanh(x)coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-32tanh^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-256tanh^{3}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-256tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh(x)coth^{3}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-16tanh(x)coth(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-32tanh(x)coth^{3}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{20tanh(x)coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{10sech^{12}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-100tanh^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-20tanh(x)coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-4sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-40tanh(x)coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-8coth^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{32tanh(x)coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{16sech^{12}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-160tanh^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh(x)coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-128tanh^{3}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-192tanh^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{512tanh^{4}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{128tanh^{3}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh(x)coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{32tanh^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{256tanh^{3}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{64tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)coth^{3}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{4coth^{2}(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-8tanh(x)coth(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-32tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-16tanh(x)coth^{3}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-4tanh(x)coth(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-8tanh(x)coth^{3}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-4coth^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{32tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh(x)coth^{3}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/4】求函数arcsin(sinh(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arcsin(sinh(x))\right)}{dx}\\=&(\frac{(cosh(x))}{((1 - (sinh(x))^{2})^{\frac{1}{2}})})\\=&\frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}})cosh(x) + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{sinh(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{sinh(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}})sinh(x)cosh^{2}(x) + \frac{cosh(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sinh(x)*2cosh(x)sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}})sinh(x) + \frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{3sinh^{2}(x)cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sinh^{2}(x)cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3sinh^{2}(x)cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sinh^{2}(x)cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{7}{2}}})sinh^{2}(x)cosh^{3}(x) + \frac{3*2sinh(x)cosh(x)cosh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{3sinh^{2}(x)*3cosh^{2}(x)sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}})cosh^{3}(x) + \frac{3cosh^{2}(x)sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + 3(\frac{\frac{-3}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}})sinh^{2}(x)cosh(x) + \frac{3*2sinh(x)cosh(x)cosh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sinh^{2}(x)sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(-2sinh(x)cosh(x) + 0)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}})cosh(x) + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{15sinh^{3}(x)cosh^{4}(x)}{(-sinh^{2}(x) + 1)^{\frac{7}{2}}} + \frac{9sinh(x)cosh^{4}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{18sinh^{3}(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{5}{2}}} + \frac{10sinh(x)cosh^{2}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sinh^{3}(x)}{(-sinh^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sinh(x)}{(-sinh^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/4】求函数arccos(cosh(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arccos(cosh(x))\right)}{dx}\\=&(\frac{-(sinh(x))}{((1 - (cosh(x))^{2})^{\frac{1}{2}})})\\=&\frac{-sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}})sinh(x) - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-sinh^{2}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-sinh^{2}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}})sinh^{2}(x)cosh(x) - \frac{2sinh(x)cosh(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh^{2}(x)sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}})cosh(x) - \frac{sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-3sinh^{3}(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sinh(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-3sinh^{3}(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sinh(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{7}{2}}})sinh^{3}(x)cosh^{2}(x) - \frac{3*3sinh^{2}(x)cosh(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3sinh^{3}(x)*2cosh(x)sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}})sinh(x)cosh^{2}(x) - \frac{3cosh(x)cosh^{2}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sinh(x)*2cosh(x)sinh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-3}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}})sinh^{3}(x) - \frac{3sinh^{2}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2cosh(x)sinh(x) + 0)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}})sinh(x) - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{-15sinh^{4}(x)cosh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{7}{2}}} - \frac{18sinh^{2}(x)cosh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{9sinh^{4}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{5}{2}}} - \frac{3cosh^{3}(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{10sinh^{2}(x)cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cosh(x)}{(-cosh^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【3/4】求函数arctan(tanh(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arctan(tanh(x))\right)}{dx}\\=&(\frac{(sech^{2}(x))}{(1 + (tanh(x))^{2})})\\=&\frac{sech^{2}(x)}{(tanh^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{sech^{2}(x)}{(tanh^{2}(x) + 1)}\right)}{dx}\\=&(\frac{-(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{2}})sech^{2}(x) + \frac{-2sech(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)}\\=&\frac{-2tanh(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2tanh(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2tanh(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2tanh(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\right)}{dx}\\=&-2(\frac{-2(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{3}})tanh(x)sech^{4}(x) - \frac{2sech^{2}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2tanh(x)*-4sech^{3}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)^{2}} - 2(\frac{-(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{2}})tanh(x)sech^{2}(x) - \frac{2sech^{2}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)} - \frac{2tanh(x)*-2sech(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)}\\=&\frac{8tanh^{2}(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{3}} - \frac{2sech^{6}(x)}{(tanh^{2}(x) + 1)^{2}} + \frac{12tanh^{2}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2sech^{4}(x)}{(tanh^{2}(x) + 1)} + \frac{4tanh^{2}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{8tanh^{2}(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{3}} - \frac{2sech^{6}(x)}{(tanh^{2}(x) + 1)^{2}} + \frac{12tanh^{2}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{2sech^{4}(x)}{(tanh^{2}(x) + 1)} + \frac{4tanh^{2}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\right)}{dx}\\=&8(\frac{-3(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{4}})tanh^{2}(x)sech^{6}(x) + \frac{8*2tanh(x)sech^{2}(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{3}} + \frac{8tanh^{2}(x)*-6sech^{5}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)^{3}} - 2(\frac{-2(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{3}})sech^{6}(x) - \frac{2*-6sech^{5}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)^{2}} + 12(\frac{-2(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{3}})tanh^{2}(x)sech^{4}(x) + \frac{12*2tanh(x)sech^{2}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} + \frac{12tanh^{2}(x)*-4sech^{3}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)^{2}} - 2(\frac{-(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{2}})sech^{4}(x) - \frac{2*-4sech^{3}(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)} + 4(\frac{-(2tanh(x)sech^{2}(x) + 0)}{(tanh^{2}(x) + 1)^{2}})tanh^{2}(x)sech^{2}(x) + \frac{4*2tanh(x)sech^{2}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)} + \frac{4tanh^{2}(x)*-2sech(x)sech(x)tanh(x)}{(tanh^{2}(x) + 1)}\\=&\frac{-48tanh^{3}(x)sech^{8}(x)}{(tanh^{2}(x) + 1)^{4}} + \frac{24tanh(x)sech^{8}(x)}{(tanh^{2}(x) + 1)^{3}} - \frac{96tanh^{3}(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{3}} + \frac{40tanh(x)sech^{6}(x)}{(tanh^{2}(x) + 1)^{2}} - \frac{56tanh^{3}(x)sech^{4}(x)}{(tanh^{2}(x) + 1)^{2}} + \frac{16tanh(x)sech^{4}(x)}{(tanh^{2}(x) + 1)} - \frac{8tanh^{3}(x)sech^{2}(x)}{(tanh^{2}(x) + 1)}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【4/4】求函数arccot(coth(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arccot(coth(x))\right)}{dx}\\=&(\frac{(-csch^{2}(x))tanh(x)sech^{8}(x)}{(1 + (coth(x))^{2})})\\=&\frac{-tanh(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-tanh(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\right)}{dx}\\=&\frac{-(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)sech^{8}(x)csch^{2}(x)}{} + \frac{-sech^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-tanh(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-tanh(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)}\\=&\frac{-2tanh(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2tanh(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\right)}{dx}\\=&\frac{-2(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh(x)coth(x)sech^{8}(x)csch^{4}(x)}{} + \frac{-2sech^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)*-csch^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)coth(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)coth(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})sech^{10}(x)csch^{2}(x)}{} + \frac{--10sech^{9}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-sech^{10}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{8(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh^{2}(x)sech^{8}(x)csch^{2}(x)}{} + \frac{8*2tanh(x)sech^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh^{2}(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{2(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)coth(x)sech^{8}(x)csch^{2}(x)}{} + \frac{2sech^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)*-csch^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)coth(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{2tanh(x)coth(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)}\\=&\frac{-8tanh(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-4coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2tanh(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{10tanh(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh^{3}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-8tanh(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-4coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2tanh(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{10tanh(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh^{3}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2tanh(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\right)}{dx}\\=&\frac{-8(\frac{-3(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{4}})tanh(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{} + \frac{-8sech^{2}(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-8tanh(x)*-2coth(x)csch^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-8tanh(x)coth^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-8tanh(x)coth^{2}(x)sech^{8}(x)*-6csch^{5}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-4(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})coth(x)sech^{10}(x)csch^{4}(x)}{} + \frac{-4*-csch^{2}(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-4coth(x)*-10sech^{9}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-4coth(x)sech^{10}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh(x)sech^{8}(x)csch^{6}(x)}{} + \frac{2sech^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2tanh(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{2tanh(x)sech^{8}(x)*-6csch^{5}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{} + \frac{32*2tanh(x)sech^{2}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)*-csch^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)coth(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh^{2}(x)coth(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{} + \frac{8sech^{2}(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)*-2coth(x)csch^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth^{2}(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{10(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)sech^{10}(x)csch^{2}(x)}{} + \frac{10sech^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{10tanh(x)*-10sech^{9}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{10tanh(x)sech^{10}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{4(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})coth(x)sech^{10}(x)csch^{2}(x)}{} + \frac{4*-csch^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4coth(x)*-10sech^{9}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{4coth(x)sech^{10}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{16(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)sech^{10}(x)csch^{2}(x)}{} + \frac{16sech^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)*-10sech^{9}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)sech^{10}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{-64(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh^{3}(x)sech^{8}(x)csch^{2}(x)}{} + \frac{-64*3tanh^{2}(x)sech^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh^{3}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh^{3}(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{-32(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{} + \frac{-32*2tanh(x)sech^{2}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)*-csch^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)coth(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh^{2}(x)coth(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{4(\frac{-2(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{3}})tanh(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{} + \frac{4sech^{2}(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{4tanh(x)*-2coth(x)csch^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{4tanh(x)coth^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{4tanh(x)coth^{2}(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)sech^{8}(x)csch^{4}(x)}{} + \frac{-2sech^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-2tanh(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-2tanh(x)sech^{8}(x)*-4csch^{3}(x)csch(x)coth(x)}{(coth^{2}(x) + 1)} + \frac{-4(\frac{-(-2coth(x)csch^{2}(x) + 0)}{(coth^{2}(x) + 1)^{2}})tanh(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{} + \frac{-4sech^{2}(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)*-2coth(x)csch^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)coth^{2}(x)*-8sech^{7}(x)sech(x)tanh(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-4tanh(x)coth^{2}(x)sech^{8}(x)*-2csch(x)csch(x)coth(x)}{(coth^{2}(x) + 1)}\\=&\frac{-48tanh(x)coth^{3}(x)sech^{8}(x)csch^{8}(x)}{(coth^{2}(x) + 1)^{4}} + \frac{-8coth^{2}(x)sech^{10}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{16tanh(x)coth(x)sech^{8}(x)csch^{8}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{64tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{48tanh(x)coth^{3}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-16coth^{2}(x)sech^{10}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{4sech^{10}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{40tanh(x)coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32coth^{2}(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{8tanh(x)coth(x)sech^{8}(x)csch^{8}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{2sech^{10}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-16tanh^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-12tanh(x)coth(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{128tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{64tanh(x)coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-32tanh^{2}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-256tanh^{3}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-256tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{32tanh(x)coth^{3}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{-16tanh(x)coth(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-32tanh(x)coth^{3}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{20tanh(x)coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{10sech^{12}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-100tanh^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-20tanh(x)coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-4sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-40tanh(x)coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-8coth^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{32tanh(x)coth(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{16sech^{12}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-160tanh^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-32tanh(x)coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-128tanh^{3}(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-192tanh^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{512tanh^{4}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{128tanh^{3}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{-64tanh(x)coth(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{32tanh^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{256tanh^{3}(x)coth(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{64tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)coth^{3}(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{3}} + \frac{4coth^{2}(x)sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-8tanh(x)coth(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-32tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-16tanh(x)coth^{3}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-4tanh(x)coth(x)sech^{8}(x)csch^{6}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-2sech^{10}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh^{2}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{16tanh(x)coth(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)} + \frac{-8tanh(x)coth^{3}(x)sech^{8}(x)csch^{4}(x)}{(coth^{2}(x) + 1)^{2}} + \frac{-4coth^{2}(x)sech^{10}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{32tanh^{2}(x)coth^{2}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)} + \frac{8tanh(x)coth^{3}(x)sech^{8}(x)csch^{2}(x)}{(coth^{2}(x) + 1)}\\ \end{split}\end{equation} \]
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