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。
\[ \begin{equation}\begin{split}【1/1】求函数arcsin(x)arccos(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arcsin(x)arccos(x)\right)}{dx}\\=&(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})arccos(x) + arcsin(x)(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{arccos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{arccos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})arccos(x) + \frac{(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})arcsin(x) - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xarccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xarcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{xarccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xarcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xarccos(x) + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xarcsin(x) - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}arccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3x^{2}arcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 1)^{2}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{2}arccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3x^{2}arcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 1)^{2}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}arccos(x) + \frac{3*2xarccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})arccos(x) + \frac{(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}arcsin(x) - \frac{3*2xarcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} - 4(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})x - \frac{4}{(-x^{2} + 1)^{2}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})arcsin(x) - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{15x^{3}arccos(x)}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9xarccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{15x^{3}arcsin(x)}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{9xarcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{24x^{2}}{(-x^{2} + 1)^{3}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{6}{(-x^{2} + 1)^{2}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数arcsin(x)arccos(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( arcsin(x)arccos(x)\right)}{dx}\\=&(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})arccos(x) + arcsin(x)(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{arccos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{arccos(x)}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})arccos(x) + \frac{(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})arcsin(x) - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xarccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xarcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{xarccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xarcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xarccos(x) + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xarcsin(x) - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}arccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3x^{2}arcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 1)^{2}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{2}arccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{arccos(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3x^{2}arcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{4x}{(-x^{2} + 1)^{2}} - \frac{arcsin(x)}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}arccos(x) + \frac{3*2xarccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})arccos(x) + \frac{(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}arcsin(x) - \frac{3*2xarcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} - 4(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})x - \frac{4}{(-x^{2} + 1)^{2}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})arcsin(x) - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\=&\frac{15x^{3}arccos(x)}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9xarccos(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{15x^{3}arcsin(x)}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{9xarcsin(x)}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{24x^{2}}{(-x^{2} + 1)^{3}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{6}{(-x^{2} + 1)^{2}}\\ \end{split}\end{equation} \]
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