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

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