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。
\[ \begin{equation}\begin{split}【1/2】求函数\frac{2ln(x)}{(\frac{3cos(x)}{arcsin(x)})} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{\frac{2}{3}ln(x)arcsin(x)}{cos(x)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{\frac{2}{3}ln(x)arcsin(x)}{cos(x)}\right)}{dx}\\=&\frac{\frac{2}{3}arcsin(x)}{(x)cos(x)} + \frac{\frac{2}{3}ln(x)sin(x)arcsin(x)}{cos^{2}(x)} + \frac{\frac{2}{3}ln(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{cos(x)}\\=&\frac{2arcsin(x)}{3xcos(x)} + \frac{2ln(x)sin(x)arcsin(x)}{3cos^{2}(x)} + \frac{2ln(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{2arcsin(x)}{3xcos(x)} + \frac{2ln(x)sin(x)arcsin(x)}{3cos^{2}(x)} + \frac{2ln(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos(x)}\right)}{dx}\\=&\frac{2*-arcsin(x)}{3x^{2}cos(x)} + \frac{2sin(x)arcsin(x)}{3xcos^{2}(x)} + \frac{2(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{3xcos(x)} + \frac{2sin(x)arcsin(x)}{3(x)cos^{2}(x)} + \frac{2ln(x)cos(x)arcsin(x)}{3cos^{2}(x)} + \frac{2ln(x)sin(x)*2sin(x)arcsin(x)}{3cos^{3}(x)} + \frac{2ln(x)sin(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{3cos^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})ln(x)}{3cos(x)} + \frac{2}{3(-x^{2} + 1)^{\frac{1}{2}}(x)cos(x)} + \frac{2ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)}\\=&\frac{-2arcsin(x)}{3x^{2}cos(x)} + \frac{4sin(x)arcsin(x)}{3xcos^{2}(x)} + \frac{2}{3(-x^{2} + 1)^{\frac{1}{2}}xcos(x)} + \frac{2ln(x)arcsin(x)}{3cos(x)} + \frac{4ln(x)sin^{2}(x)arcsin(x)}{3cos^{3}(x)} + \frac{2ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{2xln(x)}{3(-x^{2} + 1)^{\frac{3}{2}}cos(x)} + \frac{2}{3(-x^{2} + 1)^{\frac{1}{2}}xcos(x)} + \frac{2ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2arcsin(x)}{3x^{2}cos(x)} + \frac{4sin(x)arcsin(x)}{3xcos^{2}(x)} + \frac{2}{3(-x^{2} + 1)^{\frac{1}{2}}xcos(x)} + \frac{2ln(x)arcsin(x)}{3cos(x)} + \frac{4ln(x)sin^{2}(x)arcsin(x)}{3cos^{3}(x)} + \frac{2ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{2xln(x)}{3(-x^{2} + 1)^{\frac{3}{2}}cos(x)} + \frac{2}{3(-x^{2} + 1)^{\frac{1}{2}}xcos(x)} + \frac{2ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)}\right)}{dx}\\=&\frac{-2*-2arcsin(x)}{3x^{3}cos(x)} - \frac{2sin(x)arcsin(x)}{3x^{2}cos^{2}(x)} - \frac{2(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{3x^{2}cos(x)} + \frac{4*-sin(x)arcsin(x)}{3x^{2}cos^{2}(x)} + \frac{4cos(x)arcsin(x)}{3xcos^{2}(x)} + \frac{4sin(x)*2sin(x)arcsin(x)}{3xcos^{3}(x)} + \frac{4sin(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{3xcos^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{3xcos(x)} + \frac{2*-1}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos(x)} + \frac{2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{2}(x)} + \frac{2arcsin(x)}{3(x)cos(x)} + \frac{2ln(x)sin(x)arcsin(x)}{3cos^{2}(x)} + \frac{2ln(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{3cos(x)} + \frac{4sin^{2}(x)arcsin(x)}{3(x)cos^{3}(x)} + \frac{4ln(x)*2sin(x)cos(x)arcsin(x)}{3cos^{3}(x)} + \frac{4ln(x)sin^{2}(x)*3sin(x)arcsin(x)}{3cos^{4}(x)} + \frac{4ln(x)sin^{2}(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{3cos^{3}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})ln(x)sin(x)}{3cos^{2}(x)} + \frac{2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}(x)cos^{2}(x)} + \frac{2ln(x)cos(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{2ln(x)sin(x)*2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{3}(x)} + \frac{2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xln(x)}{3cos(x)} + \frac{2ln(x)}{3(-x^{2} + 1)^{\frac{3}{2}}cos(x)} + \frac{2x}{3(-x^{2} + 1)^{\frac{3}{2}}(x)cos(x)} + \frac{2xln(x)sin(x)}{3(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{3xcos(x)} + \frac{2*-1}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos(x)} + \frac{2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})ln(x)sin(x)}{3cos^{2}(x)} + \frac{2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}(x)cos^{2}(x)} + \frac{2ln(x)cos(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{2ln(x)sin(x)*2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{3}(x)}\\=&\frac{4arcsin(x)}{3x^{3}cos(x)} - \frac{2sin(x)arcsin(x)}{x^{2}cos^{2}(x)} - \frac{2}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos(x)} + \frac{2arcsin(x)}{xcos(x)} + \frac{4sin^{2}(x)arcsin(x)}{xcos^{3}(x)} + \frac{4sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{3}{2}}cos(x)} - \frac{4}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos(x)} + \frac{8sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{2}(x)} + \frac{10ln(x)sin(x)arcsin(x)}{3cos^{2}(x)} + \frac{4ln(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos(x)} + \frac{4ln(x)sin^{3}(x)arcsin(x)}{cos^{4}(x)} + \frac{4ln(x)sin^{2}(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{3}(x)} + \frac{2xln(x)sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{8ln(x)sin^{2}(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{3}(x)} + \frac{2x^{2}ln(x)}{(-x^{2} + 1)^{\frac{5}{2}}cos(x)} + \frac{2ln(x)}{3(-x^{2} + 1)^{\frac{3}{2}}cos(x)} + \frac{2ln(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{4arcsin(x)}{3x^{3}cos(x)} - \frac{2sin(x)arcsin(x)}{x^{2}cos^{2}(x)} - \frac{2}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos(x)} + \frac{2arcsin(x)}{xcos(x)} + \frac{4sin^{2}(x)arcsin(x)}{xcos^{3}(x)} + \frac{4sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{3}{2}}cos(x)} - \frac{4}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos(x)} + \frac{8sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{2}(x)} + \frac{10ln(x)sin(x)arcsin(x)}{3cos^{2}(x)} + \frac{4ln(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos(x)} + \frac{4ln(x)sin^{3}(x)arcsin(x)}{cos^{4}(x)} + \frac{4ln(x)sin^{2}(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{3}(x)} + \frac{2xln(x)sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{8ln(x)sin^{2}(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{3}(x)} + \frac{2x^{2}ln(x)}{(-x^{2} + 1)^{\frac{5}{2}}cos(x)} + \frac{2ln(x)}{3(-x^{2} + 1)^{\frac{3}{2}}cos(x)} + \frac{2ln(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos(x)}\right)}{dx}\\=&\frac{4*-3arcsin(x)}{3x^{4}cos(x)} + \frac{4sin(x)arcsin(x)}{3x^{3}cos^{2}(x)} + \frac{4(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{3x^{3}cos(x)} - \frac{2*-2sin(x)arcsin(x)}{x^{3}cos^{2}(x)} - \frac{2cos(x)arcsin(x)}{x^{2}cos^{2}(x)} - \frac{2sin(x)*2sin(x)arcsin(x)}{x^{2}cos^{3}(x)} - \frac{2sin(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{2}cos^{2}(x)} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{3x^{2}cos(x)} - \frac{2*-2}{3(-x^{2} + 1)^{\frac{1}{2}}x^{3}cos(x)} - \frac{2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos^{2}(x)} + \frac{2*-arcsin(x)}{x^{2}cos(x)} + \frac{2sin(x)arcsin(x)}{xcos^{2}(x)} + \frac{2(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{xcos(x)} + \frac{4*-sin^{2}(x)arcsin(x)}{x^{2}cos^{3}(x)} + \frac{4*2sin(x)cos(x)arcsin(x)}{xcos^{3}(x)} + \frac{4sin^{2}(x)*3sin(x)arcsin(x)}{xcos^{4}(x)} + \frac{4sin^{2}(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{xcos^{3}(x)} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(x)}{3xcos^{2}(x)} + \frac{4*-sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos^{2}(x)} + \frac{4cos(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{2}(x)} + \frac{4sin(x)*2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{3}(x)} + \frac{2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})}{cos(x)} + \frac{2sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} - \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{3x^{2}cos(x)} - \frac{4*-2}{3(-x^{2} + 1)^{\frac{1}{2}}x^{3}cos(x)} - \frac{4sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos^{2}(x)} + \frac{8(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(x)}{3xcos^{2}(x)} + \frac{8*-sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos^{2}(x)} + \frac{8cos(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{2}(x)} + \frac{8sin(x)*2sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}xcos^{3}(x)} + \frac{10sin(x)arcsin(x)}{3(x)cos^{2}(x)} + \frac{10ln(x)cos(x)arcsin(x)}{3cos^{2}(x)} + \frac{10ln(x)sin(x)*2sin(x)arcsin(x)}{3cos^{3}(x)} + \frac{10ln(x)sin(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{3cos^{2}(x)} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})ln(x)}{3cos(x)} + \frac{4}{3(-x^{2} + 1)^{\frac{1}{2}}(x)cos(x)} + \frac{4ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{4sin^{3}(x)arcsin(x)}{(x)cos^{4}(x)} + \frac{4ln(x)*3sin^{2}(x)cos(x)arcsin(x)}{cos^{4}(x)} + \frac{4ln(x)sin^{3}(x)*4sin(x)arcsin(x)}{cos^{5}(x)} + \frac{4ln(x)sin^{3}(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{cos^{4}(x)} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})ln(x)sin^{2}(x)}{3cos^{3}(x)} + \frac{4sin^{2}(x)}{3(-x^{2} + 1)^{\frac{1}{2}}(x)cos^{3}(x)} + \frac{4ln(x)*2sin(x)cos(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{3}(x)} + \frac{4ln(x)sin^{2}(x)*3sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{4}(x)} + \frac{2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xln(x)sin(x)}{cos^{2}(x)} + \frac{2ln(x)sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{2xsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}(x)cos^{2}(x)} + \frac{2xln(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{2xln(x)sin(x)*2sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos^{3}(x)} + \frac{8(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})ln(x)sin^{2}(x)}{3cos^{3}(x)} + \frac{8sin^{2}(x)}{3(-x^{2} + 1)^{\frac{1}{2}}(x)cos^{3}(x)} + \frac{8ln(x)*2sin(x)cos(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{3}(x)} + \frac{8ln(x)sin^{2}(x)*3sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{4}(x)} + \frac{2(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}ln(x)}{cos(x)} + \frac{2*2xln(x)}{(-x^{2} + 1)^{\frac{5}{2}}cos(x)} + \frac{2x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}(x)cos(x)} + \frac{2x^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{5}{2}}cos^{2}(x)} + \frac{2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})ln(x)}{3cos(x)} + \frac{2}{3(-x^{2} + 1)^{\frac{3}{2}}(x)cos(x)} + \frac{2ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})ln(x)}{3cos(x)} + \frac{2}{3(-x^{2} + 1)^{\frac{1}{2}}(x)cos(x)} + \frac{2ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)}\\=&\frac{-4arcsin(x)}{x^{4}cos(x)} + \frac{16sin(x)arcsin(x)}{3x^{3}cos^{2}(x)} + \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}xcos(x)} - \frac{4arcsin(x)}{x^{2}cos(x)} - \frac{8sin^{2}(x)arcsin(x)}{x^{2}cos^{3}(x)} - \frac{2sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos^{2}(x)} - \frac{4}{3(-x^{2} + 1)^{\frac{3}{2}}xcos(x)} - \frac{6sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}cos^{2}(x)} + \frac{40sin(x)arcsin(x)}{3xcos^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}xcos(x)} + \frac{16sin^{3}(x)arcsin(x)}{xcos^{4}(x)} + \frac{4sin^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}xcos^{3}(x)} + \frac{8sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{4}{3(-x^{2} + 1)^{\frac{1}{2}}x^{3}cos(x)} + \frac{12sin^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}xcos^{3}(x)} + \frac{8x}{(-x^{2} + 1)^{\frac{5}{2}}cos(x)} + \frac{4}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}cos(x)} + \frac{10ln(x)arcsin(x)}{3cos(x)} + \frac{56ln(x)sin^{2}(x)arcsin(x)}{3cos^{3}(x)} + \frac{10ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{4xln(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos(x)} + \frac{10ln(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{16ln(x)sin^{4}(x)arcsin(x)}{cos^{5}(x)} + \frac{4ln(x)sin^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}cos^{4}(x)} + \frac{8xln(x)sin^{2}(x)}{(-x^{2} + 1)^{\frac{3}{2}}cos^{3}(x)} + \frac{12ln(x)sin^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}cos^{4}(x)} + \frac{8x^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{5}{2}}cos^{2}(x)} + \frac{8ln(x)sin(x)}{3(-x^{2} + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{10x^{3}ln(x)}{(-x^{2} + 1)^{\frac{7}{2}}cos(x)} + \frac{6xln(x)}{(-x^{2} + 1)^{\frac{5}{2}}cos(x)}\\ \end{split}\end{equation} \]

\[ \begin{equation}\begin{split}【2/2】求函数frac(2ln(x))(frac(3cos(x))(arcsin(x))) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)arcsin(x)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)arcsin(x)\right)}{dx}\\=&\frac{6f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{(x)} + 6f^{2}r^{2}a^{2}c^{2}ln(x)*-sin(x)arcsin(x) + 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{6f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x} - 6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)arcsin(x) + \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x} - 6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)arcsin(x) + \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{6f^{2}r^{2}a^{2}c^{2}*-cos(x)arcsin(x)}{x^{2}} + \frac{6f^{2}r^{2}a^{2}c^{2}*-sin(x)arcsin(x)}{x} + \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x} - \frac{6f^{2}r^{2}a^{2}c^{2}sin(x)arcsin(x)}{(x)} - 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)arcsin(x) - 6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}) + 6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}ln(x)cos(x) + \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(x)} + \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-6f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x^{2}} - \frac{12f^{2}r^{2}a^{2}c^{2}sin(x)arcsin(x)}{x} + \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} - 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)arcsin(x) - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{6f^{2}r^{2}a^{2}c^{2}xln(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-6f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x^{2}} - \frac{12f^{2}r^{2}a^{2}c^{2}sin(x)arcsin(x)}{x} + \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} - 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)arcsin(x) - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{6f^{2}r^{2}a^{2}c^{2}xln(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{-6f^{2}r^{2}a^{2}c^{2}*-2cos(x)arcsin(x)}{x^{3}} - \frac{6f^{2}r^{2}a^{2}c^{2}*-sin(x)arcsin(x)}{x^{2}} - \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{2}} - \frac{12f^{2}r^{2}a^{2}c^{2}*-sin(x)arcsin(x)}{x^{2}} - \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x} - \frac{12f^{2}r^{2}a^{2}c^{2}sin(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x} + \frac{6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}cos(x)}{x} + \frac{6f^{2}r^{2}a^{2}c^{2}*-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} + \frac{6f^{2}r^{2}a^{2}c^{2}*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} - \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{(x)} - 6f^{2}r^{2}a^{2}c^{2}ln(x)*-sin(x)arcsin(x) - 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}) - 6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}ln(x)sin(x) - \frac{6f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}(x)} - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} + 6(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})f^{2}r^{2}a^{2}c^{2}xln(x)cos(x) + \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{6f^{2}r^{2}a^{2}c^{2}xcos(x)}{(-x^{2} + 1)^{\frac{3}{2}}(x)} + \frac{6f^{2}r^{2}a^{2}c^{2}xln(x)*-sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}cos(x)}{x} + \frac{6f^{2}r^{2}a^{2}c^{2}*-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} + \frac{6f^{2}r^{2}a^{2}c^{2}*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} - 6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}ln(x)sin(x) - \frac{6f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}(x)} - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{12f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x^{3}} + \frac{18f^{2}r^{2}a^{2}c^{2}sin(x)arcsin(x)}{x^{2}} - \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{18f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x} - \frac{12f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} + \frac{18f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{24f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} + 6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)arcsin(x) - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{18f^{2}r^{2}a^{2}c^{2}xln(x)sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{12f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{18f^{2}r^{2}a^{2}c^{2}x^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x^{3}} + \frac{18f^{2}r^{2}a^{2}c^{2}sin(x)arcsin(x)}{x^{2}} - \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{18f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x} - \frac{12f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} + \frac{18f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{24f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} + 6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)arcsin(x) - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{18f^{2}r^{2}a^{2}c^{2}xln(x)sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{12f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{18f^{2}r^{2}a^{2}c^{2}x^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{12f^{2}r^{2}a^{2}c^{2}*-3cos(x)arcsin(x)}{x^{4}} + \frac{12f^{2}r^{2}a^{2}c^{2}*-sin(x)arcsin(x)}{x^{3}} + \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{3}} + \frac{18f^{2}r^{2}a^{2}c^{2}*-2sin(x)arcsin(x)}{x^{3}} + \frac{18f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x^{2}} + \frac{18f^{2}r^{2}a^{2}c^{2}sin(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{2}} - \frac{6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}cos(x)}{x^{2}} - \frac{6f^{2}r^{2}a^{2}c^{2}*-2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}} - \frac{6f^{2}r^{2}a^{2}c^{2}*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{18f^{2}r^{2}a^{2}c^{2}*-cos(x)arcsin(x)}{x^{2}} - \frac{18f^{2}r^{2}a^{2}c^{2}*-sin(x)arcsin(x)}{x} - \frac{18f^{2}r^{2}a^{2}c^{2}cos(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x} - \frac{12(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}sin(x)}{x} - \frac{12f^{2}r^{2}a^{2}c^{2}*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} + 18(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})f^{2}r^{2}a^{2}c^{2}cos(x) + \frac{18f^{2}r^{2}a^{2}c^{2}*-sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{12(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}cos(x)}{x^{2}} - \frac{12f^{2}r^{2}a^{2}c^{2}*-2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}} - \frac{12f^{2}r^{2}a^{2}c^{2}*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{24(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}sin(x)}{x} - \frac{24f^{2}r^{2}a^{2}c^{2}*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{24f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} + \frac{6f^{2}r^{2}a^{2}c^{2}sin(x)arcsin(x)}{(x)} + 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)arcsin(x) + 6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}) - 6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}ln(x)cos(x) - \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(x)} - \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - 18(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})f^{2}r^{2}a^{2}c^{2}xln(x)sin(x) - \frac{18f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{18f^{2}r^{2}a^{2}c^{2}xsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}(x)} - \frac{18f^{2}r^{2}a^{2}c^{2}xln(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - 12(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})f^{2}r^{2}a^{2}c^{2}ln(x)cos(x) - \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(x)} - \frac{12f^{2}r^{2}a^{2}c^{2}ln(x)*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}} + 18(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})f^{2}r^{2}a^{2}c^{2}x^{2}ln(x)cos(x) + \frac{18f^{2}r^{2}a^{2}c^{2}*2xln(x)cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{18f^{2}r^{2}a^{2}c^{2}x^{2}cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}(x)} + \frac{18f^{2}r^{2}a^{2}c^{2}x^{2}ln(x)*-sin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + 6(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})f^{2}r^{2}a^{2}c^{2}ln(x)cos(x) + \frac{6f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}(x)} + \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)*-sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{-36f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x^{4}} - \frac{48f^{2}r^{2}a^{2}c^{2}sin(x)arcsin(x)}{x^{3}} + \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}} + \frac{36f^{2}r^{2}a^{2}c^{2}cos(x)arcsin(x)}{x^{2}} + \frac{18f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} - \frac{12f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}x} + \frac{36f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}} + \frac{54f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}} + \frac{24f^{2}r^{2}a^{2}c^{2}sin(x)arcsin(x)}{x} - \frac{18f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} - \frac{72f^{2}r^{2}a^{2}c^{2}sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{54f^{2}r^{2}a^{2}c^{2}cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}x} + \frac{72f^{2}r^{2}a^{2}c^{2}xcos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + 6f^{2}r^{2}a^{2}c^{2}ln(x)cos(x)arcsin(x) + \frac{6f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{36f^{2}r^{2}a^{2}c^{2}xln(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{18f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{72f^{2}r^{2}a^{2}c^{2}x^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{24f^{2}r^{2}a^{2}c^{2}ln(x)sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{90f^{2}r^{2}a^{2}c^{2}x^{3}ln(x)cos(x)}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{54f^{2}r^{2}a^{2}c^{2}xln(x)cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]

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