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

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