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

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