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

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