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

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