There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ sin(\frac{arcsin(x)}{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = sin(\frac{1}{2}arcsin(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( sin(\frac{1}{2}arcsin(x))\right)}{dx}\\=&cos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(\frac{1}{2}arcsin(x))}{2} + \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcos(\frac{1}{2}arcsin(x))}{2} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x*-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{cos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3x^{2}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}cos(\frac{1}{2}arcsin(x))}{2} + \frac{3*2xcos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}*-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{5}{2}}} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})cos(\frac{1}{2}arcsin(x))}{2} + \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xsin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xcos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xsin(\frac{1}{2}arcsin(x))}{2} - \frac{sin(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{2}} - \frac{xcos(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{2(-x^{2} + 1)^{2}} - \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)} - \frac{-sin(\frac{1}{2}arcsin(x))*\frac{1}{2}(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{8(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}cos(\frac{1}{2}arcsin(x))}{2(-x^{2} + 1)^{\frac{7}{2}}} + \frac{33xcos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{3x^{2}sin(\frac{1}{2}arcsin(x))}{(-x^{2} + 1)^{3}} - \frac{3sin(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{2}} - \frac{xcos(\frac{1}{2}arcsin(x))}{8(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{xcos(\frac{1}{2}arcsin(x))}{4(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{sin(\frac{1}{2}arcsin(x))}{16(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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