求导函数：
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
输入一个原函数（即需要求导的函数），然后设置需要求导的变量和求导的阶数，点击“下一步”按钮，即可获得该函数相应阶数的导函数。
注意，输入的函数支持数学函数和其它常量。
本次共计算 1 个题目：每一题对 x 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数log_{sin(x)}^{arcsin(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( log_{sin(x)}^{arcsin(x)}\right)}{dx}\\=&(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})\\=&\frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin(x)} - \frac{log_{sin(x)}^{arcsin(x)}cos(x)}{ln(sin(x))sin(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin(x)} - \frac{log_{sin(x)}^{arcsin(x)}cos(x)}{ln(sin(x))sin(x)}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{ln(sin(x))arcsin(x)} + \frac{-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))(sin(x))arcsin(x)} + \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))} - \frac{(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos(x)}{ln(sin(x))sin(x)} - \frac{log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln^{2}(sin(x))(sin(x))sin(x)} - \frac{log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln(sin(x))sin^{2}(x)} - \frac{log_{sin(x)}^{arcsin(x)}*-sin(x)}{ln(sin(x))sin(x)}\\=&\frac{x}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} + \frac{2log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}}{ln(sin(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} + \frac{2log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}}{ln(sin(x))}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{ln(sin(x))arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} + \frac{x*-cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))(sin(x))arcsin(x)} + \frac{x(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin(x)arcsin(x)} - \frac{-cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{cos(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))(sin(x))arcsin^{2}(x)} - \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin(x)arcsin(x)} - \frac{-cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{cos(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)} + \frac{2(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{2log_{sin(x)}^{arcsin(x)}*-2cos(x)cos^{2}(x)}{ln^{3}(sin(x))(sin(x))sin^{2}(x)} + \frac{2log_{sin(x)}^{arcsin(x)}*-2cos(x)cos^{2}(x)}{ln^{2}(sin(x))sin^{3}(x)} + \frac{2log_{sin(x)}^{arcsin(x)}*-2cos(x)sin(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{2}(x)}{ln(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}*-cos(x)cos^{2}(x)}{ln^{2}(sin(x))(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}*-2cos(x)cos^{2}(x)}{ln(sin(x))sin^{3}(x)} + \frac{log_{sin(x)}^{arcsin(x)}*-2cos(x)sin(x)}{ln(sin(x))sin^{2}(x)} + \frac{(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})}{ln(sin(x))} + \frac{log_{sin(x)}^{arcsin(x)}*-cos(x)}{ln^{2}(sin(x))(sin(x))}\\=&\frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} - \frac{3xcos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} + \frac{4cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin(x)} + \frac{2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} - \frac{2x}{(-x^{2} + 1)^{2}ln(sin(x))arcsin^{2}(x)} + \frac{cos(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{3}(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos(x)}{ln^{2}(sin(x))sin(x)} + \frac{cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{2log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln(sin(x))sin^{3}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}cos(x)}{ln(sin(x))sin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} - \frac{3xcos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} + \frac{4cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin(x)} + \frac{2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} - \frac{2x}{(-x^{2} + 1)^{2}ln(sin(x))arcsin^{2}(x)} + \frac{cos(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{3}(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos(x)}{ln^{2}(sin(x))sin(x)} + \frac{cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{2log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln(sin(x))sin^{3}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}cos(x)}{ln(sin(x))sin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin(x)}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}}{ln(sin(x))arcsin(x)} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin(x)} + \frac{3x^{2}*-cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}ln^{2}(sin(x))(sin(x))arcsin(x)} + \frac{3x^{2}(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})}{ln(sin(x))arcsin(x)} + \frac{-cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))(sin(x))arcsin(x)} + \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))} - \frac{3(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcos(x)}{ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3x*-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{3}(sin(x))(sin(x))sin(x)arcsin(x)} - \frac{3x*-cos(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{3x*-sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3xcos(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{x*-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))(sin(x))arcsin^{2}(x)} - \frac{x(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{4*-3cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{4}(sin(x))(sin(x))sin^{2}(x)arcsin(x)} + \frac{4*-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{3}(x)arcsin(x)} + \frac{4*-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{4cos^{2}(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2*-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin^{2}(x)arcsin(x)} + \frac{2*-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{3}(x)arcsin(x)} + \frac{2*-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2cos^{2}(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{ln^{2}(sin(x))arcsin(x)} + \frac{2*-2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))arcsin(x)} + \frac{2(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2*-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin(x)arcsin^{2}(x)} + \frac{2*-cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin^{2}(x)} + \frac{2*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2cos(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)} - \frac{2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})x}{ln(sin(x))arcsin^{2}(x)} - \frac{2}{(-x^{2} + 1)^{2}ln(sin(x))arcsin^{2}(x)} - \frac{2x*-cos(x)}{(-x^{2} + 1)^{2}ln^{2}(sin(x))(sin(x))arcsin^{2}(x)} - \frac{2x(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}ln(sin(x))} + \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})cos(x)}{ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{-2cos(x)cos(x)}{(-x^{2} + 1)ln^{3}(sin(x))(sin(x))sin(x)arcsin^{2}(x)} + \frac{-cos(x)cos(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin^{2}(x)arcsin^{2}(x)} + \frac{-sin(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{cos(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)ln^{2}(sin(x))sin(x)} + \frac{2(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{3}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)ln(sin(x))arcsin^{3}(x)} + \frac{2*-cos(x)}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))(sin(x))arcsin^{3}(x)} + \frac{2(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2*-3cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{4}(sin(x))(sin(x))sin^{2}(x)arcsin(x)} + \frac{2*-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{3}(x)arcsin(x)} + \frac{2*-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2cos^{2}(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)} - \frac{6(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{3}(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos(x)cos^{3}(x)}{ln^{4}(sin(x))(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos(x)cos^{3}(x)}{ln^{3}(sin(x))sin^{4}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos^{2}(x)sin(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{3}(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-2cos(x)cos^{3}(x)}{ln^{3}(sin(x))(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos(x)cos^{3}(x)}{ln^{2}(sin(x))sin^{4}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos^{2}(x)sin(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{6(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos(x)}{ln^{2}(sin(x))sin(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-2cos(x)cos(x)}{ln^{3}(sin(x))(sin(x))sin(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-sin(x)}{ln^{2}(sin(x))sin(x)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin^{2}(x)arcsin(x)} + \frac{-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{3}(x)arcsin(x)} + \frac{-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{cos^{2}(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)} - \frac{2(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{3}(x)}{ln(sin(x))sin^{3}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-cos(x)cos^{3}(x)}{ln^{2}(sin(x))(sin(x))sin^{3}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-3cos(x)cos^{3}(x)}{ln(sin(x))sin^{4}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-3cos^{2}(x)sin(x)}{ln(sin(x))sin^{3}(x)} - \frac{2(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos(x)}{ln(sin(x))sin(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln^{2}(sin(x))(sin(x))sin(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln(sin(x))sin^{2}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-sin(x)}{ln(sin(x))sin(x)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{ln^{2}(sin(x))arcsin(x)} + \frac{-2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))arcsin(x)} + \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))}\\=&\frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}ln(sin(x))arcsin(x)} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin(x)} - \frac{12x^{2}cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{4cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} + \frac{12xcos^{2}(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{6xcos^{2}(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{6x}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))arcsin(x)} + \frac{3xcos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} - \frac{12x^{2}}{(-x^{2} + 1)^{3}ln(sin(x))arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)^{2}ln(sin(x))arcsin^{2}(x)} + \frac{9xcos(x)}{(-x^{2} + 1)^{2}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2x}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{3}(x)} - \frac{18cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{4}(sin(x))sin^{3}(x)arcsin(x)} - \frac{18cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{3}(x)arcsin(x)} - \frac{18cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin(x)arcsin(x)} - \frac{6cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin^{2}(x)} - \frac{6cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{3}(x)arcsin(x)} - \frac{6cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin^{2}(x)} - \frac{6cos^{2}(x)}{(-x^{2} + 1)ln^{3}(sin(x))sin^{2}(x)arcsin^{2}(x)} - \frac{3cos^{2}(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin^{2}(x)arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)ln^{2}(sin(x))arcsin^{2}(x)} - \frac{4cos(x)}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{3}(x)} + \frac{4x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}ln(sin(x))arcsin^{3}(x)} - \frac{2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)ln^{2}(sin(x))sin(x)arcsin^{3}(x)} + \frac{6x}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin^{3}(x)} - \frac{2cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin^{3}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{4}(x)} - \frac{6cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{4}(sin(x))sin^{3}(x)arcsin(x)} + \frac{24log_{sin(x)}^{arcsin(x)}cos^{4}(x)}{ln^{4}(sin(x))sin^{4}(x)} + \frac{36log_{sin(x)}^{arcsin(x)}cos^{4}(x)}{ln^{3}(sin(x))sin^{4}(x)} + \frac{36log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)} - \frac{6cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{3}(x)arcsin(x)} + \frac{22log_{sin(x)}^{arcsin(x)}cos^{4}(x)}{ln^{2}(sin(x))sin^{4}(x)} + \frac{28log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin(x)arcsin(x)} + \frac{8log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln(sin(x))sin^{2}(x)} - \frac{2cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{3}(x)arcsin(x)} + \frac{6log_{sin(x)}^{arcsin(x)}cos^{4}(x)}{ln(sin(x))sin^{4}(x)} + \frac{6log_{sin(x)}^{arcsin(x)}}{ln^{2}(sin(x))} - \frac{2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} + \frac{2log_{sin(x)}^{arcsin(x)}}{ln(sin(x))}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数log_{sin(x)}^{arcsin(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( log_{sin(x)}^{arcsin(x)}\right)}{dx}\\=&(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})\\=&\frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin(x)} - \frac{log_{sin(x)}^{arcsin(x)}cos(x)}{ln(sin(x))sin(x)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin(x)} - \frac{log_{sin(x)}^{arcsin(x)}cos(x)}{ln(sin(x))sin(x)}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{ln(sin(x))arcsin(x)} + \frac{-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))(sin(x))arcsin(x)} + \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))} - \frac{(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos(x)}{ln(sin(x))sin(x)} - \frac{log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln^{2}(sin(x))(sin(x))sin(x)} - \frac{log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln(sin(x))sin^{2}(x)} - \frac{log_{sin(x)}^{arcsin(x)}*-sin(x)}{ln(sin(x))sin(x)}\\=&\frac{x}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} + \frac{2log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}}{ln(sin(x))}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} + \frac{2log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}}{ln(sin(x))}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{ln(sin(x))arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} + \frac{x*-cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))(sin(x))arcsin(x)} + \frac{x(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin(x)arcsin(x)} - \frac{-cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{cos(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))(sin(x))arcsin^{2}(x)} - \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin(x)arcsin(x)} - \frac{-cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{cos(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)} + \frac{2(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{2log_{sin(x)}^{arcsin(x)}*-2cos(x)cos^{2}(x)}{ln^{3}(sin(x))(sin(x))sin^{2}(x)} + \frac{2log_{sin(x)}^{arcsin(x)}*-2cos(x)cos^{2}(x)}{ln^{2}(sin(x))sin^{3}(x)} + \frac{2log_{sin(x)}^{arcsin(x)}*-2cos(x)sin(x)}{ln^{2}(sin(x))sin^{2}(x)} + \frac{(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{2}(x)}{ln(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}*-cos(x)cos^{2}(x)}{ln^{2}(sin(x))(sin(x))sin^{2}(x)} + \frac{log_{sin(x)}^{arcsin(x)}*-2cos(x)cos^{2}(x)}{ln(sin(x))sin^{3}(x)} + \frac{log_{sin(x)}^{arcsin(x)}*-2cos(x)sin(x)}{ln(sin(x))sin^{2}(x)} + \frac{(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})}{ln(sin(x))} + \frac{log_{sin(x)}^{arcsin(x)}*-cos(x)}{ln^{2}(sin(x))(sin(x))}\\=&\frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} - \frac{3xcos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} + \frac{4cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin(x)} + \frac{2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} - \frac{2x}{(-x^{2} + 1)^{2}ln(sin(x))arcsin^{2}(x)} + \frac{cos(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{3}(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos(x)}{ln^{2}(sin(x))sin(x)} + \frac{cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{2log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln(sin(x))sin^{3}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}cos(x)}{ln(sin(x))sin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin(x)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin(x)} - \frac{3xcos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} + \frac{4cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin(x)} + \frac{2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} - \frac{2x}{(-x^{2} + 1)^{2}ln(sin(x))arcsin^{2}(x)} + \frac{cos(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{3}(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}cos(x)}{ln^{2}(sin(x))sin(x)} + \frac{cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{2log_{sin(x)}^{arcsin(x)}cos^{3}(x)}{ln(sin(x))sin^{3}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}cos(x)}{ln(sin(x))sin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin(x)}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}}{ln(sin(x))arcsin(x)} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin(x)} + \frac{3x^{2}*-cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}ln^{2}(sin(x))(sin(x))arcsin(x)} + \frac{3x^{2}(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})}{ln(sin(x))arcsin(x)} + \frac{-cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))(sin(x))arcsin(x)} + \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))} - \frac{3(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcos(x)}{ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3x*-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{3}(sin(x))(sin(x))sin(x)arcsin(x)} - \frac{3x*-cos(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} - \frac{3x*-sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3xcos(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{x*-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))(sin(x))arcsin^{2}(x)} - \frac{x(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{4*-3cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{4}(sin(x))(sin(x))sin^{2}(x)arcsin(x)} + \frac{4*-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{3}(x)arcsin(x)} + \frac{4*-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{4cos^{2}(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2*-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin^{2}(x)arcsin(x)} + \frac{2*-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{3}(x)arcsin(x)} + \frac{2*-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2cos^{2}(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{ln^{2}(sin(x))arcsin(x)} + \frac{2*-2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))arcsin(x)} + \frac{2(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2*-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin(x)arcsin^{2}(x)} + \frac{2*-cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin^{2}(x)} + \frac{2*-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2cos(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)} - \frac{2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})x}{ln(sin(x))arcsin^{2}(x)} - \frac{2}{(-x^{2} + 1)^{2}ln(sin(x))arcsin^{2}(x)} - \frac{2x*-cos(x)}{(-x^{2} + 1)^{2}ln^{2}(sin(x))(sin(x))arcsin^{2}(x)} - \frac{2x(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}ln(sin(x))} + \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})cos(x)}{ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{-2cos(x)cos(x)}{(-x^{2} + 1)ln^{3}(sin(x))(sin(x))sin(x)arcsin^{2}(x)} + \frac{-cos(x)cos(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin^{2}(x)arcsin^{2}(x)} + \frac{-sin(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{cos(x)(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)ln^{2}(sin(x))sin(x)} + \frac{2(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})}{(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{3}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)ln(sin(x))arcsin^{3}(x)} + \frac{2*-cos(x)}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))(sin(x))arcsin^{3}(x)} + \frac{2(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2*-3cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{4}(sin(x))(sin(x))sin^{2}(x)arcsin(x)} + \frac{2*-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{3}(x)arcsin(x)} + \frac{2*-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{2cos^{2}(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)} - \frac{6(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{3}(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos(x)cos^{3}(x)}{ln^{4}(sin(x))(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos(x)cos^{3}(x)}{ln^{3}(sin(x))sin^{4}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos^{2}(x)sin(x)}{ln^{3}(sin(x))sin^{3}(x)} - \frac{6(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{3}(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-2cos(x)cos^{3}(x)}{ln^{3}(sin(x))(sin(x))sin^{3}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos(x)cos^{3}(x)}{ln^{2}(sin(x))sin^{4}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-3cos^{2}(x)sin(x)}{ln^{2}(sin(x))sin^{3}(x)} - \frac{6(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos(x)}{ln^{2}(sin(x))sin(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-2cos(x)cos(x)}{ln^{3}(sin(x))(sin(x))sin(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6log_{sin(x)}^{arcsin(x)}*-sin(x)}{ln^{2}(sin(x))sin(x)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))sin^{2}(x)arcsin(x)} + \frac{-2cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{3}(x)arcsin(x)} + \frac{-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{cos^{2}(x)(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)} - \frac{2(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos^{3}(x)}{ln(sin(x))sin^{3}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-cos(x)cos^{3}(x)}{ln^{2}(sin(x))(sin(x))sin^{3}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-3cos(x)cos^{3}(x)}{ln(sin(x))sin^{4}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-3cos^{2}(x)sin(x)}{ln(sin(x))sin^{3}(x)} - \frac{2(\frac{(\frac{((\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})}))}{(arcsin(x))} - \frac{(cos(x))log_{sin(x)}^{arcsin(x)}}{(sin(x))})}{(ln(sin(x)))})cos(x)}{ln(sin(x))sin(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln^{2}(sin(x))(sin(x))sin(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-cos(x)cos(x)}{ln(sin(x))sin^{2}(x)} - \frac{2log_{sin(x)}^{arcsin(x)}*-sin(x)}{ln(sin(x))sin(x)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{ln^{2}(sin(x))arcsin(x)} + \frac{-2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))(sin(x))arcsin(x)} + \frac{(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))}\\=&\frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}ln(sin(x))arcsin(x)} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin(x)} - \frac{12x^{2}cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin^{2}(x)} - \frac{4cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln(sin(x))arcsin^{2}(x)} + \frac{12xcos^{2}(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin(x)} + \frac{6xcos^{2}(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin(x)} + \frac{6x}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))arcsin(x)} + \frac{3xcos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} - \frac{12x^{2}}{(-x^{2} + 1)^{3}ln(sin(x))arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)^{2}ln(sin(x))arcsin^{2}(x)} + \frac{9xcos(x)}{(-x^{2} + 1)^{2}ln^{2}(sin(x))sin(x)arcsin^{2}(x)} + \frac{2x}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{3}(x)} - \frac{18cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{4}(sin(x))sin^{3}(x)arcsin(x)} - \frac{18cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{3}(x)arcsin(x)} - \frac{18cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin(x)arcsin(x)} - \frac{6cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{2}(x)arcsin^{2}(x)} - \frac{6cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{3}(x)arcsin(x)} - \frac{6cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} - \frac{3cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{2}(x)arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))arcsin^{2}(x)} - \frac{6cos^{2}(x)}{(-x^{2} + 1)ln^{3}(sin(x))sin^{2}(x)arcsin^{2}(x)} - \frac{3cos^{2}(x)}{(-x^{2} + 1)ln^{2}(sin(x))sin^{2}(x)arcsin^{2}(x)} - \frac{3}{(-x^{2} + 1)ln^{2}(sin(x))arcsin^{2}(x)} - \frac{4cos(x)}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin^{3}(x)} + \frac{4x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}ln(sin(x))arcsin^{3}(x)} - \frac{2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)ln^{2}(sin(x))sin(x)arcsin^{3}(x)} + \frac{6x}{(-x^{2} + 1)^{\frac{5}{2}}ln(sin(x))arcsin^{3}(x)} - \frac{2cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}(sin(x))sin(x)arcsin^{3}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}ln(sin(x))arcsin^{4}(x)} - \frac{6cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{4}(sin(x))sin^{3}(x)arcsin(x)} + \frac{24log_{sin(x)}^{arcsin(x)}cos^{4}(x)}{ln^{4}(sin(x))sin^{4}(x)} + \frac{36log_{sin(x)}^{arcsin(x)}cos^{4}(x)}{ln^{3}(sin(x))sin^{4}(x)} + \frac{36log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln^{3}(sin(x))sin^{2}(x)} - \frac{6cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin^{3}(x)arcsin(x)} + \frac{22log_{sin(x)}^{arcsin(x)}cos^{4}(x)}{ln^{2}(sin(x))sin^{4}(x)} + \frac{28log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln^{2}(sin(x))sin^{2}(x)} - \frac{6cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{3}(sin(x))sin(x)arcsin(x)} + \frac{8log_{sin(x)}^{arcsin(x)}cos^{2}(x)}{ln(sin(x))sin^{2}(x)} - \frac{2cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin^{3}(x)arcsin(x)} + \frac{6log_{sin(x)}^{arcsin(x)}cos^{4}(x)}{ln(sin(x))sin^{4}(x)} + \frac{6log_{sin(x)}^{arcsin(x)}}{ln^{2}(sin(x))} - \frac{2cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}ln^{2}(sin(x))sin(x)arcsin(x)} + \frac{2log_{sin(x)}^{arcsin(x)}}{ln(sin(x))}\\ \end{split}\end{equation} \]
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