There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ (e^{2x}arctan(sqrt(e^{x} - 1)))*2 - (16)(e^{x} - 1)(32) - (12)(sqrt(e^{x} - 1))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = 2e^{2x}arctan(sqrt(e^{x} - 1)) - 512e^{x} - 12sqrt(e^{x} - 1) + 512\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 2e^{2x}arctan(sqrt(e^{x} - 1)) - 512e^{x} - 12sqrt(e^{x} - 1) + 512\right)}{dx}\\=&2e^{2x}*2arctan(sqrt(e^{x} - 1)) + 2e^{2x}(\frac{(\frac{(e^{x} + 0)*\frac{1}{2}}{(e^{x} - 1)^{\frac{1}{2}}})}{(1 + (sqrt(e^{x} - 1))^{2})}) - 512e^{x} - \frac{12(e^{x} + 0)*\frac{1}{2}}{(e^{x} - 1)^{\frac{1}{2}}} + 0\\=&4e^{2x}arctan(sqrt(e^{x} - 1)) + \frac{e^{2x}e^{x}}{(e^{x} - 1)^{\frac{1}{2}}(sqrt(e^{x} - 1)^{2} + 1)} - 512e^{x} - \frac{6e^{x}}{(e^{x} - 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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