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 3 个题目：每一题对 x 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/3】求函数e^{arcsin(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{arcsin(x)}\right)}{dx}\\=&e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arcsin(x)}}{(-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{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arcsin(x)}}{(-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}}})xe^{arcsin(x)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(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}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-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}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arcsin(x)} + \frac{3*2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}(\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}}})e^{arcsin(x)} + \frac{e^{arcsin(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}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arcsin(x)} + \frac{2e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{2xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{12xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{12x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{3}} + \frac{3e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/3】求函数e^{arccos(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{arccos(x)}\right)}{dx}\\=&e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{-e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)} - \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-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{-xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-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}}})xe^{arccos(x)} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xe^{arccos(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}}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arccos(x)} - \frac{3*2xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}e^{arccos(x)}(\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}}})e^{arccos(x)} - \frac{e^{arccos(x)}(\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}}})xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{xe^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arccos(x)} + \frac{2e^{arccos(x)}}{(-x^{2} + 1)^{2}} + \frac{2xe^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)}}{(-x^{2} + 1)} - \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\\=&\frac{-15x^{3}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{12xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{12x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{3}} + \frac{3e^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} - \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【3/3】求函数e^{arctan(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{arctan(x)}\right)}{dx}\\=&e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})\\=&\frac{e^{arctan(x)}}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{e^{arctan(x)}}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)}\\=&\frac{-2xe^{arctan(x)}}{(x^{2} + 1)^{2}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2xe^{arctan(x)}}{(x^{2} + 1)^{2}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xe^{arctan(x)} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{2xe^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + (\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}}\\=&\frac{8x^{2}e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{6xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{8x^{2}e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{6xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{3}}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}e^{arctan(x)} + \frac{8*2xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{8x^{2}e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})e^{arctan(x)} - \frac{2e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - 6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})xe^{arctan(x)} - \frac{6e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{6xe^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + (\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}}\\=&\frac{-48x^{3}e^{arctan(x)}}{(x^{2} + 1)^{4}} + \frac{24xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{44x^{2}e^{arctan(x)}}{(x^{2} + 1)^{4}} - \frac{8e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{12xe^{arctan(x)}}{(x^{2} + 1)^{4}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{4}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/3】求函数e^{arcsin(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{arcsin(x)}\right)}{dx}\\=&e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arcsin(x)}}{(-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{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arcsin(x)}}{(-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}}})xe^{arcsin(x)} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(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}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-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}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arcsin(x)} + \frac{3*2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}(\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}}})e^{arcsin(x)} + \frac{e^{arcsin(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}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arcsin(x)} + \frac{2e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{2xe^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arcsin(x)}}{(-x^{2} + 1)} + \frac{e^{arcsin(x)}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{12xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{12x^{2}e^{arcsin(x)}}{(-x^{2} + 1)^{3}} + \frac{3e^{arcsin(x)}}{(-x^{2} + 1)^{2}} + \frac{xe^{arcsin(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arcsin(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{e^{arcsin(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/3】求函数e^{arccos(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{arccos(x)}\right)}{dx}\\=&e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{-e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)} - \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-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{-xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-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}}})xe^{arccos(x)} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xe^{arccos(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}}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}e^{arccos(x)} - \frac{3*2xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}e^{arccos(x)}(\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}}})e^{arccos(x)} - \frac{e^{arccos(x)}(\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}}})xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{xe^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xe^{arccos(x)} + \frac{2e^{arccos(x)}}{(-x^{2} + 1)^{2}} + \frac{2xe^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})e^{arccos(x)}}{(-x^{2} + 1)} - \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{e^{arccos(x)}(\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)}\\=&\frac{-15x^{3}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{12xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{\frac{5}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{12x^{2}e^{arccos(x)}}{(-x^{2} + 1)^{3}} + \frac{3e^{arccos(x)}}{(-x^{2} + 1)^{2}} - \frac{xe^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} - \frac{2xe^{arccos(x)}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{e^{arccos(x)}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【3/3】求函数e^{arctan(x)} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( e^{arctan(x)}\right)}{dx}\\=&e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})\\=&\frac{e^{arctan(x)}}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{e^{arctan(x)}}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)}\\=&\frac{-2xe^{arctan(x)}}{(x^{2} + 1)^{2}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-2xe^{arctan(x)}}{(x^{2} + 1)^{2}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xe^{arctan(x)} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{2xe^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + (\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}}\\=&\frac{8x^{2}e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{6xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{3}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{8x^{2}e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{2e^{arctan(x)}}{(x^{2} + 1)^{2}} - \frac{6xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{3}}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}e^{arctan(x)} + \frac{8*2xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{8x^{2}e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})e^{arctan(x)} - \frac{2e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - 6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})xe^{arctan(x)} - \frac{6e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{6xe^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + (\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})e^{arctan(x)} + \frac{e^{arctan(x)}(\frac{(1)}{(1 + (x)^{2})})}{(x^{2} + 1)^{3}}\\=&\frac{-48x^{3}e^{arctan(x)}}{(x^{2} + 1)^{4}} + \frac{24xe^{arctan(x)}}{(x^{2} + 1)^{3}} + \frac{44x^{2}e^{arctan(x)}}{(x^{2} + 1)^{4}} - \frac{8e^{arctan(x)}}{(x^{2} + 1)^{3}} - \frac{12xe^{arctan(x)}}{(x^{2} + 1)^{4}} + \frac{e^{arctan(x)}}{(x^{2} + 1)^{4}}\\ \end{split}\end{equation} \]
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