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

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