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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{sin(x)}{(1 + {x}^{2})} 关于 x 的 3 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{sin(x)}{(x^{2} + 1)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{sin(x)}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})sin(x) + \frac{cos(x)}{(x^{2} + 1)}\\=&\frac{-2xsin(x)}{(x^{2} + 1)^{2}} + \frac{cos(x)}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-2xsin(x)}{(x^{2} + 1)^{2}} + \frac{cos(x)}{(x^{2} + 1)}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xsin(x) - \frac{2sin(x)}{(x^{2} + 1)^{2}} - \frac{2xcos(x)}{(x^{2} + 1)^{2}} + (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})cos(x) + \frac{-sin(x)}{(x^{2} + 1)}\\=&\frac{8x^{2}sin(x)}{(x^{2} + 1)^{3}} - \frac{2sin(x)}{(x^{2} + 1)^{2}} - \frac{4xcos(x)}{(x^{2} + 1)^{2}} - \frac{sin(x)}{(x^{2} + 1)}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{8x^{2}sin(x)}{(x^{2} + 1)^{3}} - \frac{2sin(x)}{(x^{2} + 1)^{2}} - \frac{4xcos(x)}{(x^{2} + 1)^{2}} - \frac{sin(x)}{(x^{2} + 1)}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}sin(x) + \frac{8*2xsin(x)}{(x^{2} + 1)^{3}} + \frac{8x^{2}cos(x)}{(x^{2} + 1)^{3}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})sin(x) - \frac{2cos(x)}{(x^{2} + 1)^{2}} - 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xcos(x) - \frac{4cos(x)}{(x^{2} + 1)^{2}} - \frac{4x*-sin(x)}{(x^{2} + 1)^{2}} - (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})sin(x) - \frac{cos(x)}{(x^{2} + 1)}\\=&\frac{-48x^{3}sin(x)}{(x^{2} + 1)^{4}} + \frac{24xsin(x)}{(x^{2} + 1)^{3}} + \frac{24x^{2}cos(x)}{(x^{2} + 1)^{3}} - \frac{6cos(x)}{(x^{2} + 1)^{2}} + \frac{6xsin(x)}{(x^{2} + 1)^{2}} - \frac{cos(x)}{(x^{2} + 1)}\\ \end{split}\end{equation} \]

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