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

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