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

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