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 7 个题目：每一题对 x 求 4 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/7】求函数sin(x) - arcsin(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(x) - arcsin(x)\right)}{dx}\\=&cos(x) - (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&cos(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( cos(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-sin(x) - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-sin(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -sin(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&-cos(x) - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&-cos(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -cos(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} - \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&sin(x) - \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/7】求函数sin(x) - arccos(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(x) - arccos(x)\right)}{dx}\\=&cos(x) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&cos(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( cos(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-sin(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-sin(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -sin(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&-cos(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&-cos(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -cos(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&sin(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【3/7】求函数cos(x) - arcsin(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cos(x) - arcsin(x)\right)}{dx}\\=&-sin(x) - (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&-sin(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -sin(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-cos(x) - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-cos(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -cos(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&sin(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( sin(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&cos(x) - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} - \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&cos(x) - \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【4/7】求函数cos(x) - arccos(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cos(x) - arccos(x)\right)}{dx}\\=&-sin(x) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&-sin(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -sin(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-cos(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-cos(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -cos(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&sin(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( sin(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&cos(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&cos(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【5/7】求函数cos(x) - arctan(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cos(x) - arctan(x)\right)}{dx}\\=&-sin(x) - (\frac{(1)}{(1 + (x)^{2})})\\=&-sin(x) - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -sin(x) - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&-cos(x) - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&-cos(x) + \frac{2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -cos(x) + \frac{2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&--sin(x) + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{2}{(x^{2} + 1)^{2}}\\=&sin(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( sin(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&cos(x) - 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{8*2x}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&cos(x) + \frac{48x^{3}}{(x^{2} + 1)^{4}} - \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【6/7】求函数tan(x) - arccos(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( tan(x) - arccos(x)\right)}{dx}\\=&sec^{2}(x)(1) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&sec^{2}(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( sec^{2}(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&2sec^{2}(x)tan(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&2tan(x)sec^{2}(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2tan(x)sec^{2}(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【7/7】求函数tan(x) - arctan(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( tan(x) - arctan(x)\right)}{dx}\\=&sec^{2}(x)(1) - (\frac{(1)}{(1 + (x)^{2})})\\=&sec^{2}(x) - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( sec^{2}(x) - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&2sec^{2}(x)tan(x) - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&2tan(x)sec^{2}(x) + \frac{2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2tan(x)sec^{2}(x) + \frac{2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{2}{(x^{2} + 1)^{2}}\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) - 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{8*2x}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) + \frac{48x^{3}}{(x^{2} + 1)^{4}} - \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
>>注：本次最多计算 7 道题。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/7】求函数sin(x) - arcsin(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(x) - arcsin(x)\right)}{dx}\\=&cos(x) - (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&cos(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( cos(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-sin(x) - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-sin(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -sin(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&-cos(x) - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&-cos(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -cos(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} - \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&sin(x) - \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【2/7】求函数sin(x) - arccos(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(x) - arccos(x)\right)}{dx}\\=&cos(x) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&cos(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( cos(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-sin(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-sin(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -sin(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&-cos(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&-cos(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( -cos(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&sin(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【3/7】求函数cos(x) - arcsin(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cos(x) - arcsin(x)\right)}{dx}\\=&-sin(x) - (\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&-sin(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -sin(x) - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-cos(x) - (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-cos(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -cos(x) - \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&sin(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( sin(x) - \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&cos(x) - 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} - \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&cos(x) - \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【4/7】求函数cos(x) - arccos(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cos(x) - arccos(x)\right)}{dx}\\=&-sin(x) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&-sin(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -sin(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-cos(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&-cos(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -cos(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&--sin(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&sin(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( sin(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&cos(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&cos(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【5/7】求函数cos(x) - arctan(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( cos(x) - arctan(x)\right)}{dx}\\=&-sin(x) - (\frac{(1)}{(1 + (x)^{2})})\\=&-sin(x) - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( -sin(x) - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&-cos(x) - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&-cos(x) + \frac{2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( -cos(x) + \frac{2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&--sin(x) + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{2}{(x^{2} + 1)^{2}}\\=&sin(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( sin(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&cos(x) - 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{8*2x}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&cos(x) + \frac{48x^{3}}{(x^{2} + 1)^{4}} - \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【6/7】求函数tan(x) - arccos(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( tan(x) - arccos(x)\right)}{dx}\\=&sec^{2}(x)(1) - (\frac{-(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&sec^{2}(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( sec^{2}(x) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&2sec^{2}(x)tan(x) + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})\\=&2tan(x)sec^{2}(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2tan(x)sec^{2}(x) + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]
\[ \begin{equation}\begin{split}【7/7】求函数tan(x) - arctan(x) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( tan(x) - arctan(x)\right)}{dx}\\=&sec^{2}(x)(1) - (\frac{(1)}{(1 + (x)^{2})})\\=&sec^{2}(x) - \frac{1}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( sec^{2}(x) - \frac{1}{(x^{2} + 1)}\right)}{dx}\\=&2sec^{2}(x)tan(x) - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})\\=&2tan(x)sec^{2}(x) + \frac{2x}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( 2tan(x)sec^{2}(x) + \frac{2x}{(x^{2} + 1)^{2}}\right)}{dx}\\=&2sec^{2}(x)(1)sec^{2}(x) + 2tan(x)*2sec^{2}(x)tan(x) + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x + \frac{2}{(x^{2} + 1)^{2}}\\=&2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( 2sec^{4}(x) + 4tan^{2}(x)sec^{2}(x) - \frac{8x^{2}}{(x^{2} + 1)^{3}} + \frac{2}{(x^{2} + 1)^{2}}\right)}{dx}\\=&2*4sec^{4}(x)tan(x) + 4*2tan(x)sec^{2}(x)(1)sec^{2}(x) + 4tan^{2}(x)*2sec^{2}(x)tan(x) - 8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2} - \frac{8*2x}{(x^{2} + 1)^{3}} + 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})\\=&16tan(x)sec^{4}(x) + 8tan^{3}(x)sec^{2}(x) + \frac{48x^{3}}{(x^{2} + 1)^{4}} - \frac{24x}{(x^{2} + 1)^{3}}\\ \end{split}\end{equation} \]
>>注：本次最多计算 7 道题。
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