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

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