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

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