There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ 2e^{x}({(e^{x} - 1)}^{\frac{1}{2}}) - (\frac{4}{3})({(e^{x} - 1)}^{\frac{3}{2}})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = 2(e^{x} - 1)^{\frac{1}{2}}e^{x} - \frac{4}{3}(e^{x} - 1)^{\frac{3}{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 2(e^{x} - 1)^{\frac{1}{2}}e^{x} - \frac{4}{3}(e^{x} - 1)^{\frac{3}{2}}\right)}{dx}\\=&2(\frac{\frac{1}{2}(e^{x} + 0)}{(e^{x} - 1)^{\frac{1}{2}}})e^{x} + 2(e^{x} - 1)^{\frac{1}{2}}e^{x} - \frac{4}{3}(\frac{3}{2}(e^{x} - 1)^{\frac{1}{2}}(e^{x} + 0))\\=&\frac{e^{{x}*{2}}}{(e^{x} - 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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