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

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