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。
\[ \begin{equation}\begin{split}【1/1】求函数\frac{({(x + 5)}^{2})(\frac{{(x - 4)}^{1}}{3})}{(({(x + 2)}^{5})(\frac{{(x + 4)}^{1}}{2}))} 关于 x 的 1 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &原函数 = \frac{\frac{2}{3}x^{3}}{(x + 2)^{5}(x + 4)} + \frac{4x^{2}}{(x + 2)^{5}(x + 4)} - \frac{10x}{(x + 2)^{5}(x + 4)} - \frac{\frac{200}{3}}{(x + 2)^{5}(x + 4)}\\&\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( \frac{\frac{2}{3}x^{3}}{(x + 2)^{5}(x + 4)} + \frac{4x^{2}}{(x + 2)^{5}(x + 4)} - \frac{10x}{(x + 2)^{5}(x + 4)} - \frac{\frac{200}{3}}{(x + 2)^{5}(x + 4)}\right)}{dx}\\=&\frac{\frac{2}{3}(\frac{-5(1 + 0)}{(x + 2)^{6}})x^{3}}{(x + 4)} + \frac{\frac{2}{3}(\frac{-(1 + 0)}{(x + 4)^{2}})x^{3}}{(x + 2)^{5}} + \frac{\frac{2}{3}*3x^{2}}{(x + 2)^{5}(x + 4)} + \frac{4(\frac{-5(1 + 0)}{(x + 2)^{6}})x^{2}}{(x + 4)} + \frac{4(\frac{-(1 + 0)}{(x + 4)^{2}})x^{2}}{(x + 2)^{5}} + \frac{4*2x}{(x + 2)^{5}(x + 4)} - \frac{10(\frac{-5(1 + 0)}{(x + 2)^{6}})x}{(x + 4)} - \frac{10(\frac{-(1 + 0)}{(x + 4)^{2}})x}{(x + 2)^{5}} - \frac{10}{(x + 2)^{5}(x + 4)} - \frac{\frac{200}{3}(\frac{-5(1 + 0)}{(x + 2)^{6}})}{(x + 4)} - \frac{\frac{200}{3}(\frac{-(1 + 0)}{(x + 4)^{2}})}{(x + 2)^{5}}\\=&\frac{-10x^{3}}{3(x + 2)^{6}(x + 4)} - \frac{2x^{3}}{3(x + 4)^{2}(x + 2)^{5}} + \frac{2x^{2}}{(x + 4)(x + 2)^{5}} - \frac{20x^{2}}{(x + 2)^{6}(x + 4)} - \frac{4x^{2}}{(x + 4)^{2}(x + 2)^{5}} + \frac{8x}{(x + 4)(x + 2)^{5}} + \frac{50x}{(x + 2)^{6}(x + 4)} + \frac{10x}{(x + 4)^{2}(x + 2)^{5}} + \frac{200}{3(x + 4)^{2}(x + 2)^{5}} - \frac{10}{(x + 2)^{5}(x + 4)} + \frac{1000}{3(x + 2)^{6}(x + 4)}\\ \end{split}\end{equation} \]

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