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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{x}{({x}^{2} + {a}^{2})} 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{x}{(x^{2} + a^{2})}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{x}{(x^{2} + a^{2})}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + a^{2})^{2}})x + \frac{1}{(x^{2} + a^{2})}\\=&\frac{-2x^{2}}{(x^{2} + a^{2})^{2}} + \frac{1}{(x^{2} + a^{2})}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{-2x^{2}}{(x^{2} + a^{2})^{2}} + \frac{1}{(x^{2} + a^{2})}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + a^{2})^{3}})x^{2} - \frac{2*2x}{(x^{2} + a^{2})^{2}} + (\frac{-(2x + 0)}{(x^{2} + a^{2})^{2}})\\=&\frac{8x^{3}}{(x^{2} + a^{2})^{3}} - \frac{6x}{(x^{2} + a^{2})^{2}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{8x^{3}}{(x^{2} + a^{2})^{3}} - \frac{6x}{(x^{2} + a^{2})^{2}}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + a^{2})^{4}})x^{3} + \frac{8*3x^{2}}{(x^{2} + a^{2})^{3}} - 6(\frac{-2(2x + 0)}{(x^{2} + a^{2})^{3}})x - \frac{6}{(x^{2} + a^{2})^{2}}\\=&\frac{-48x^{4}}{(x^{2} + a^{2})^{4}} + \frac{48x^{2}}{(x^{2} + a^{2})^{3}} - \frac{6}{(x^{2} + a^{2})^{2}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{-48x^{4}}{(x^{2} + a^{2})^{4}} + \frac{48x^{2}}{(x^{2} + a^{2})^{3}} - \frac{6}{(x^{2} + a^{2})^{2}}\right)}{dx}\\=&-48(\frac{-4(2x + 0)}{(x^{2} + a^{2})^{5}})x^{4} - \frac{48*4x^{3}}{(x^{2} + a^{2})^{4}} + 48(\frac{-3(2x + 0)}{(x^{2} + a^{2})^{4}})x^{2} + \frac{48*2x}{(x^{2} + a^{2})^{3}} - 6(\frac{-2(2x + 0)}{(x^{2} + a^{2})^{3}})\\=&\frac{384x^{5}}{(x^{2} + a^{2})^{5}} - \frac{480x^{3}}{(x^{2} + a^{2})^{4}} + \frac{120x}{(x^{2} + a^{2})^{3}}\\ \end{split}\end{equation} \]

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