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

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