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。
\[ \begin{equation}\begin{split}【1/1】求函数sin(\frac{arcsin(x)}{2}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = sin(\frac{1}{2}arcsin(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(\frac{1}{2}arcsin(x))\right)}{dx}\\=&cos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(\frac{1}{2}arcsin(x))}{2} + \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-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{xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcos(\frac{1}{2}arcsin(x))}{2} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x*-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{cos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{2}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}cos(\frac{1}{2}arcsin(x))}{2} + \frac{3*2xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}*-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})cos(\frac{1}{2}arcsin(x))}{2} + \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xcos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xsin(\frac{1}{2}arcsin(x))}{2} - \frac{sin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{xcos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{2}} - \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)} - \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{7}{2}}} + \frac{33xcos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{3x^{2}sin(\frac{1}{2}arcsin(x))}{(-x^{2} + 1)^{3}} - \frac{3sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{2}} - \frac{xcos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xcos(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{sin(\frac{1}{2}arcsin(x))}{16(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数sin(\frac{arcsin(x)}{2}) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = sin(\frac{1}{2}arcsin(x))\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(\frac{1}{2}arcsin(x))\right)}{dx}\\=&cos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(\frac{1}{2}arcsin(x))}{2} + \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-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{xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcos(\frac{1}{2}arcsin(x))}{2} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x*-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{cos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{2}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}cos(\frac{1}{2}arcsin(x))}{2} + \frac{3*2xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}*-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})cos(\frac{1}{2}arcsin(x))}{2} + \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xcos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xsin(\frac{1}{2}arcsin(x))}{2} - \frac{sin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{xcos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{2}} - \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)} - \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{7}{2}}} + \frac{33xcos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{3x^{2}sin(\frac{1}{2}arcsin(x))}{(-x^{2} + 1)^{3}} - \frac{3sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{2}} - \frac{xcos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xcos(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{sin(\frac{1}{2}arcsin(x))}{16(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
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