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

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