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 3 阶导数。
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数S*2 - 2({x}^{3}cos(\frac{x}{2}) + \frac{1}{2})sqrt(4 - {x}^{2})dx 关于 x 的 3 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = 2S - 2dx^{4}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - dxsqrt(-x^{2} + 4)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( 2S - 2dx^{4}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - dxsqrt(-x^{2} + 4)\right)}{dx}\\=&0 - 2d*4x^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - 2dx^{4}*-sin(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) - \frac{2dx^{4}cos(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - dsqrt(-x^{2} + 4) - \frac{dx(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}}\\=& - 8dx^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) + dx^{4}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{2dx^{5}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} - dsqrt(-x^{2} + 4) + \frac{dx^{2}}{(-x^{2} + 4)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( - 8dx^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) + dx^{4}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{2dx^{5}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} - dsqrt(-x^{2} + 4) + \frac{dx^{2}}{(-x^{2} + 4)^{\frac{1}{2}}}\right)}{dx}\\=& - 8d*3x^{2}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - 8dx^{3}*-sin(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) - \frac{8dx^{3}cos(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + d*4x^{3}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + dx^{4}cos(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) + \frac{dx^{4}sin(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx^{5}cos(\frac{1}{2}x) + \frac{2d*5x^{4}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{2dx^{5}*-sin(\frac{1}{2}x)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{d(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx^{2} + \frac{d*2x}{(-x^{2} + 4)^{\frac{1}{2}}}\\=& - 24dx^{2}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) + 8dx^{3}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{18dx^{4}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{dx^{4}cos(\frac{1}{2}x)sqrt(-x^{2} + 4)}{2} - \frac{2dx^{5}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{2dx^{6}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{dx^{3}}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{3dx}{(-x^{2} + 4)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - 24dx^{2}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) + 8dx^{3}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{18dx^{4}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{dx^{4}cos(\frac{1}{2}x)sqrt(-x^{2} + 4)}{2} - \frac{2dx^{5}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{2dx^{6}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{dx^{3}}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{3dx}{(-x^{2} + 4)^{\frac{1}{2}}}\right)}{dx}\\=& - 24d*2xcos(\frac{1}{2}x)sqrt(-x^{2} + 4) - 24dx^{2}*-sin(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) - \frac{24dx^{2}cos(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 8d*3x^{2}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + 8dx^{3}cos(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) + \frac{8dx^{3}sin(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 18(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx^{4}cos(\frac{1}{2}x) + \frac{18d*4x^{3}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{18dx^{4}*-sin(\frac{1}{2}x)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{d*4x^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4)}{2} + \frac{dx^{4}*-sin(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4)}{2} + \frac{dx^{4}cos(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{2(-x^{2} + 4)^{\frac{1}{2}}} - 2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx^{5}sin(\frac{1}{2}x) - \frac{2d*5x^{4}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{2dx^{5}cos(\frac{1}{2}x)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})dx^{6}cos(\frac{1}{2}x) + \frac{2d*6x^{5}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{2dx^{6}*-sin(\frac{1}{2}x)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{3}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})dx^{3} + \frac{d*3x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}} + 3(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx + \frac{3d}{(-x^{2} + 4)^{\frac{1}{2}}}\\=& - 48dxcos(\frac{1}{2}x)sqrt(-x^{2} + 4) + 36dx^{2}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{96dx^{3}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + 6dx^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - \frac{27dx^{4}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{30dx^{5}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{dx^{4}sin(\frac{1}{2}x)sqrt(-x^{2} + 4)}{4} - \frac{3dx^{5}cos(\frac{1}{2}x)}{2(-x^{2} + 4)^{\frac{1}{2}}} - \frac{3dx^{6}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6dx^{7}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{5}{2}}} + \frac{3dx^{4}}{(-x^{2} + 4)^{\frac{5}{2}}} + \frac{6dx^{2}}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{3d}{(-x^{2} + 4)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数S*2 - 2({x}^{3}cos(\frac{x}{2}) + \frac{1}{2})sqrt(4 - {x}^{2})dx 关于 x 的 3 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = 2S - 2dx^{4}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - dxsqrt(-x^{2} + 4)\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( 2S - 2dx^{4}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - dxsqrt(-x^{2} + 4)\right)}{dx}\\=&0 - 2d*4x^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - 2dx^{4}*-sin(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) - \frac{2dx^{4}cos(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - dsqrt(-x^{2} + 4) - \frac{dx(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}}\\=& - 8dx^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) + dx^{4}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{2dx^{5}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} - dsqrt(-x^{2} + 4) + \frac{dx^{2}}{(-x^{2} + 4)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( - 8dx^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) + dx^{4}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{2dx^{5}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} - dsqrt(-x^{2} + 4) + \frac{dx^{2}}{(-x^{2} + 4)^{\frac{1}{2}}}\right)}{dx}\\=& - 8d*3x^{2}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - 8dx^{3}*-sin(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) - \frac{8dx^{3}cos(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + d*4x^{3}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + dx^{4}cos(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) + \frac{dx^{4}sin(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx^{5}cos(\frac{1}{2}x) + \frac{2d*5x^{4}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{2dx^{5}*-sin(\frac{1}{2}x)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{d(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx^{2} + \frac{d*2x}{(-x^{2} + 4)^{\frac{1}{2}}}\\=& - 24dx^{2}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) + 8dx^{3}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{18dx^{4}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{dx^{4}cos(\frac{1}{2}x)sqrt(-x^{2} + 4)}{2} - \frac{2dx^{5}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{2dx^{6}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{dx^{3}}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{3dx}{(-x^{2} + 4)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( - 24dx^{2}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) + 8dx^{3}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{18dx^{4}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{dx^{4}cos(\frac{1}{2}x)sqrt(-x^{2} + 4)}{2} - \frac{2dx^{5}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{2dx^{6}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{dx^{3}}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{3dx}{(-x^{2} + 4)^{\frac{1}{2}}}\right)}{dx}\\=& - 24d*2xcos(\frac{1}{2}x)sqrt(-x^{2} + 4) - 24dx^{2}*-sin(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) - \frac{24dx^{2}cos(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 8d*3x^{2}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + 8dx^{3}cos(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4) + \frac{8dx^{3}sin(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 18(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx^{4}cos(\frac{1}{2}x) + \frac{18d*4x^{3}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{18dx^{4}*-sin(\frac{1}{2}x)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{d*4x^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4)}{2} + \frac{dx^{4}*-sin(\frac{1}{2}x)*\frac{1}{2}sqrt(-x^{2} + 4)}{2} + \frac{dx^{4}cos(\frac{1}{2}x)(-2x + 0)*\frac{1}{2}}{2(-x^{2} + 4)^{\frac{1}{2}}} - 2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx^{5}sin(\frac{1}{2}x) - \frac{2d*5x^{4}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} - \frac{2dx^{5}cos(\frac{1}{2}x)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{1}{2}}} + 2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})dx^{6}cos(\frac{1}{2}x) + \frac{2d*6x^{5}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{2dx^{6}*-sin(\frac{1}{2}x)*\frac{1}{2}}{(-x^{2} + 4)^{\frac{3}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{5}{2}}})dx^{3} + \frac{d*3x^{2}}{(-x^{2} + 4)^{\frac{3}{2}}} + 3(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 4)^{\frac{3}{2}}})dx + \frac{3d}{(-x^{2} + 4)^{\frac{1}{2}}}\\=& - 48dxcos(\frac{1}{2}x)sqrt(-x^{2} + 4) + 36dx^{2}sin(\frac{1}{2}x)sqrt(-x^{2} + 4) + \frac{96dx^{3}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + 6dx^{3}cos(\frac{1}{2}x)sqrt(-x^{2} + 4) - \frac{27dx^{4}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{1}{2}}} + \frac{30dx^{5}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} - \frac{dx^{4}sin(\frac{1}{2}x)sqrt(-x^{2} + 4)}{4} - \frac{3dx^{5}cos(\frac{1}{2}x)}{2(-x^{2} + 4)^{\frac{1}{2}}} - \frac{3dx^{6}sin(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{6dx^{7}cos(\frac{1}{2}x)}{(-x^{2} + 4)^{\frac{5}{2}}} + \frac{3dx^{4}}{(-x^{2} + 4)^{\frac{5}{2}}} + \frac{6dx^{2}}{(-x^{2} + 4)^{\frac{3}{2}}} + \frac{3d}{(-x^{2} + 4)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]
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