本次共计算 1 个题目：每一题对 x 求 1 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数{(\frac{({x}^{2} - 2x)(1 - {x}^{2})}{({x}^{2} + 1)})}^{\frac{1}{2}} 关于 x 的 1 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = (\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( (\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}\right)}{dx}\\=&(\frac{\frac{1}{2}(-(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{4} - \frac{4x^{3}}{(x^{2} + 1)} + (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{2} + \frac{2x}{(x^{2} + 1)} + 2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x^{3} + \frac{2*3x^{2}}{(x^{2} + 1)} - 2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})x - \frac{2}{(x^{2} + 1)})}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}})\\=&\frac{x^{5}}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}(x^{2} + 1)^{2}} - \frac{2x^{3}}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}(x^{2} + 1)} - \frac{x^{3}}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}(x^{2} + 1)^{2}} + \frac{x}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}(x^{2} + 1)} - \frac{2x^{4}}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}(x^{2} + 1)^{2}} + \frac{3x^{2}}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}(x^{2} + 1)} + \frac{2x^{2}}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}(x^{2} + 1)^{2}} - \frac{1}{(\frac{-x^{4}}{(x^{2} + 1)} + \frac{x^{2}}{(x^{2} + 1)} + \frac{2x^{3}}{(x^{2} + 1)} - \frac{2x}{(x^{2} + 1)})^{\frac{1}{2}}(x^{2} + 1)}\\ \end{split}\end{equation} \]

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