There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ {(\frac{1}{2}x)}^{(\frac{x}{(1 - x)})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (\frac{1}{2}x)^{(\frac{x}{(-x + 1)})}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (\frac{1}{2}x)^{(\frac{x}{(-x + 1)})}\right)}{dx}\\=&((\frac{1}{2}x)^{(\frac{x}{(-x + 1)})}(((\frac{-(-1 + 0)}{(-x + 1)^{2}})x + \frac{1}{(-x + 1)})ln(\frac{1}{2}x) + \frac{(\frac{x}{(-x + 1)})(\frac{1}{2})}{(\frac{1}{2}x)}))\\=&\frac{x(\frac{1}{2}x)^{(\frac{x}{(-x + 1)})}ln(\frac{1}{2}x)}{(-x + 1)^{2}} + \frac{(\frac{1}{2}x)^{(\frac{x}{(-x + 1)})}ln(\frac{1}{2}x)}{(-x + 1)} + \frac{(\frac{1}{2}x)^{(\frac{x}{(-x + 1)})}}{(-x + 1)}\\ \end{split}\end{equation} \]

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