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。
\[ \begin{equation}\begin{split}【1/1】求函数{(xx + 3x + 3)}^{\frac{1}{2}} - {(xx + 3)}^{\frac{1}{2}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = (x^{2} + 3x + 3)^{\frac{1}{2}} - (x^{2} + 3)^{\frac{1}{2}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( (x^{2} + 3x + 3)^{\frac{1}{2}} - (x^{2} + 3)^{\frac{1}{2}}\right)}{dx}\\=&(\frac{\frac{1}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{1}{2}}}) - (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 3)^{\frac{1}{2}}})\\=&\frac{x}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - \frac{x}{(x^{2} + 3)^{\frac{1}{2}}} + \frac{3}{2(x^{2} + 3x + 3)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - \frac{x}{(x^{2} + 3)^{\frac{1}{2}}} + \frac{3}{2(x^{2} + 3x + 3)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{3}{2}}})x + \frac{1}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - (\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 3)^{\frac{3}{2}}})x - \frac{1}{(x^{2} + 3)^{\frac{1}{2}}} + \frac{3(\frac{\frac{-1}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{3}{2}}})}{2}\\=&\frac{-x^{2}}{(x^{2} + 3x + 3)^{\frac{3}{2}}} - \frac{3x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{x^{2}}{(x^{2} + 3)^{\frac{3}{2}}} + \frac{1}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - \frac{1}{(x^{2} + 3)^{\frac{1}{2}}} - \frac{9}{4(x^{2} + 3x + 3)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-x^{2}}{(x^{2} + 3x + 3)^{\frac{3}{2}}} - \frac{3x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{x^{2}}{(x^{2} + 3)^{\frac{3}{2}}} + \frac{1}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - \frac{1}{(x^{2} + 3)^{\frac{1}{2}}} - \frac{9}{4(x^{2} + 3x + 3)^{\frac{3}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})x^{2} - \frac{2x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} - 3(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})x - \frac{3}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + (\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 3)^{\frac{5}{2}}})x^{2} + \frac{2x}{(x^{2} + 3)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{3}{2}}}) - (\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 3)^{\frac{3}{2}}}) - \frac{9(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})}{4}\\=&\frac{3x^{3}}{(x^{2} + 3x + 3)^{\frac{5}{2}}} + \frac{27x^{2}}{2(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{3x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81x}{4(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{3x^{3}}{(x^{2} + 3)^{\frac{5}{2}}} + \frac{3x}{(x^{2} + 3)^{\frac{3}{2}}} - \frac{9}{2(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81}{8(x^{2} + 3x + 3)^{\frac{5}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{3}}{(x^{2} + 3x + 3)^{\frac{5}{2}}} + \frac{27x^{2}}{2(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{3x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81x}{4(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{3x^{3}}{(x^{2} + 3)^{\frac{5}{2}}} + \frac{3x}{(x^{2} + 3)^{\frac{3}{2}}} - \frac{9}{2(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81}{8(x^{2} + 3x + 3)^{\frac{5}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{7}{2}}})x^{3} + \frac{3*3x^{2}}{(x^{2} + 3x + 3)^{\frac{5}{2}}} + \frac{27(\frac{\frac{-5}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{7}{2}}})x^{2}}{2} + \frac{27*2x}{2(x^{2} + 3x + 3)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})x - \frac{3}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81(\frac{\frac{-5}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{7}{2}}})x}{4} + \frac{81}{4(x^{2} + 3x + 3)^{\frac{5}{2}}} - 3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 3)^{\frac{7}{2}}})x^{3} - \frac{3*3x^{2}}{(x^{2} + 3)^{\frac{5}{2}}} + 3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 3)^{\frac{5}{2}}})x + \frac{3}{(x^{2} + 3)^{\frac{3}{2}}} - \frac{9(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})}{2} + \frac{81(\frac{\frac{-5}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{7}{2}}})}{8}\\=&\frac{-15x^{4}}{(x^{2} + 3x + 3)^{\frac{7}{2}}} - \frac{90x^{3}}{(x^{2} + 3x + 3)^{\frac{7}{2}}} + \frac{18x^{2}}{(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{405x^{2}}{2(x^{2} + 3x + 3)^{\frac{7}{2}}} + \frac{54x}{(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{18x^{2}}{(x^{2} + 3)^{\frac{5}{2}}} - \frac{405x}{2(x^{2} + 3x + 3)^{\frac{7}{2}}} + \frac{15x^{4}}{(x^{2} + 3)^{\frac{7}{2}}} - \frac{3}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81}{2(x^{2} + 3x + 3)^{\frac{5}{2}}} + \frac{3}{(x^{2} + 3)^{\frac{3}{2}}} - \frac{1215}{16(x^{2} + 3x + 3)^{\frac{7}{2}}}\\ \end{split}\end{equation} \]
注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数{(xx + 3x + 3)}^{\frac{1}{2}} - {(xx + 3)}^{\frac{1}{2}} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = (x^{2} + 3x + 3)^{\frac{1}{2}} - (x^{2} + 3)^{\frac{1}{2}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( (x^{2} + 3x + 3)^{\frac{1}{2}} - (x^{2} + 3)^{\frac{1}{2}}\right)}{dx}\\=&(\frac{\frac{1}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{1}{2}}}) - (\frac{\frac{1}{2}(2x + 0)}{(x^{2} + 3)^{\frac{1}{2}}})\\=&\frac{x}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - \frac{x}{(x^{2} + 3)^{\frac{1}{2}}} + \frac{3}{2(x^{2} + 3x + 3)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{x}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - \frac{x}{(x^{2} + 3)^{\frac{1}{2}}} + \frac{3}{2(x^{2} + 3x + 3)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{3}{2}}})x + \frac{1}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - (\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 3)^{\frac{3}{2}}})x - \frac{1}{(x^{2} + 3)^{\frac{1}{2}}} + \frac{3(\frac{\frac{-1}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{3}{2}}})}{2}\\=&\frac{-x^{2}}{(x^{2} + 3x + 3)^{\frac{3}{2}}} - \frac{3x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{x^{2}}{(x^{2} + 3)^{\frac{3}{2}}} + \frac{1}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - \frac{1}{(x^{2} + 3)^{\frac{1}{2}}} - \frac{9}{4(x^{2} + 3x + 3)^{\frac{3}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{-x^{2}}{(x^{2} + 3x + 3)^{\frac{3}{2}}} - \frac{3x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{x^{2}}{(x^{2} + 3)^{\frac{3}{2}}} + \frac{1}{(x^{2} + 3x + 3)^{\frac{1}{2}}} - \frac{1}{(x^{2} + 3)^{\frac{1}{2}}} - \frac{9}{4(x^{2} + 3x + 3)^{\frac{3}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})x^{2} - \frac{2x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} - 3(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})x - \frac{3}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + (\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 3)^{\frac{5}{2}}})x^{2} + \frac{2x}{(x^{2} + 3)^{\frac{3}{2}}} + (\frac{\frac{-1}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{3}{2}}}) - (\frac{\frac{-1}{2}(2x + 0)}{(x^{2} + 3)^{\frac{3}{2}}}) - \frac{9(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})}{4}\\=&\frac{3x^{3}}{(x^{2} + 3x + 3)^{\frac{5}{2}}} + \frac{27x^{2}}{2(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{3x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81x}{4(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{3x^{3}}{(x^{2} + 3)^{\frac{5}{2}}} + \frac{3x}{(x^{2} + 3)^{\frac{3}{2}}} - \frac{9}{2(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81}{8(x^{2} + 3x + 3)^{\frac{5}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{3}}{(x^{2} + 3x + 3)^{\frac{5}{2}}} + \frac{27x^{2}}{2(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{3x}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81x}{4(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{3x^{3}}{(x^{2} + 3)^{\frac{5}{2}}} + \frac{3x}{(x^{2} + 3)^{\frac{3}{2}}} - \frac{9}{2(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81}{8(x^{2} + 3x + 3)^{\frac{5}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{7}{2}}})x^{3} + \frac{3*3x^{2}}{(x^{2} + 3x + 3)^{\frac{5}{2}}} + \frac{27(\frac{\frac{-5}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{7}{2}}})x^{2}}{2} + \frac{27*2x}{2(x^{2} + 3x + 3)^{\frac{5}{2}}} - 3(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})x - \frac{3}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81(\frac{\frac{-5}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{7}{2}}})x}{4} + \frac{81}{4(x^{2} + 3x + 3)^{\frac{5}{2}}} - 3(\frac{\frac{-5}{2}(2x + 0)}{(x^{2} + 3)^{\frac{7}{2}}})x^{3} - \frac{3*3x^{2}}{(x^{2} + 3)^{\frac{5}{2}}} + 3(\frac{\frac{-3}{2}(2x + 0)}{(x^{2} + 3)^{\frac{5}{2}}})x + \frac{3}{(x^{2} + 3)^{\frac{3}{2}}} - \frac{9(\frac{\frac{-3}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{5}{2}}})}{2} + \frac{81(\frac{\frac{-5}{2}(2x + 3 + 0)}{(x^{2} + 3x + 3)^{\frac{7}{2}}})}{8}\\=&\frac{-15x^{4}}{(x^{2} + 3x + 3)^{\frac{7}{2}}} - \frac{90x^{3}}{(x^{2} + 3x + 3)^{\frac{7}{2}}} + \frac{18x^{2}}{(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{405x^{2}}{2(x^{2} + 3x + 3)^{\frac{7}{2}}} + \frac{54x}{(x^{2} + 3x + 3)^{\frac{5}{2}}} - \frac{18x^{2}}{(x^{2} + 3)^{\frac{5}{2}}} - \frac{405x}{2(x^{2} + 3x + 3)^{\frac{7}{2}}} + \frac{15x^{4}}{(x^{2} + 3)^{\frac{7}{2}}} - \frac{3}{(x^{2} + 3x + 3)^{\frac{3}{2}}} + \frac{81}{2(x^{2} + 3x + 3)^{\frac{5}{2}}} + \frac{3}{(x^{2} + 3)^{\frac{3}{2}}} - \frac{1215}{16(x^{2} + 3x + 3)^{\frac{7}{2}}}\\ \end{split}\end{equation} \]
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