There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ arcsin({(\frac{(1 + sin(x))}{2})}^{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = arcsin(\frac{1}{4}sin^{2}(x) + \frac{1}{2}sin(x) + \frac{1}{4})\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arcsin(\frac{1}{4}sin^{2}(x) + \frac{1}{2}sin(x) + \frac{1}{4})\right)}{dx}\\=&(\frac{(\frac{1}{4}*2sin(x)cos(x) + \frac{1}{2}cos(x) + 0)}{((1 - (\frac{1}{4}sin^{2}(x) + \frac{1}{2}sin(x) + \frac{1}{4})^{2})^{\frac{1}{2}})})\\=&\frac{sin(x)cos(x)}{2(\frac{-1}{16}sin^{4}(x) - \frac{1}{4}sin^{3}(x) - \frac{3}{8}sin^{2}(x) - \frac{1}{4}sin(x) + \frac{15}{16})^{\frac{1}{2}}} + \frac{cos(x)}{2(\frac{-1}{16}sin^{4}(x) - \frac{1}{4}sin^{3}(x) - \frac{3}{8}sin^{2}(x) - \frac{1}{4}sin(x) + \frac{15}{16})^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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