There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ 2arcsin(\frac{sqrt(x)}{2})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = 2arcsin(\frac{1}{2}sqrt(x))\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( 2arcsin(\frac{1}{2}sqrt(x))\right)}{dx}\\=&2(\frac{(\frac{\frac{1}{2}*\frac{1}{2}}{(x)^{\frac{1}{2}}})}{((1 - (\frac{1}{2}sqrt(x))^{2})^{\frac{1}{2}})})\\=&\frac{1}{2(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{2(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{1}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(\frac{\frac{-1}{4}*2(x)^{\frac{1}{2}}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + 0)}{(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}})}{2x^{\frac{1}{2}}} + \frac{\frac{-1}{2}}{2(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{3}{2}}}\\=&\frac{1}{16(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}x^{\frac{1}{2}}} - \frac{1}{4(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{3}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{16(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}x^{\frac{1}{2}}} - \frac{1}{4(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{3}{2}}}\right)}{dx}\\=&\frac{(\frac{\frac{-3}{2}(\frac{\frac{-1}{4}*2(x)^{\frac{1}{2}}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + 0)}{(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{5}{2}}})}{16x^{\frac{1}{2}}} + \frac{\frac{-1}{2}}{16(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}x^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(\frac{\frac{-1}{4}*2(x)^{\frac{1}{2}}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + 0)}{(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}})}{4x^{\frac{3}{2}}} - \frac{\frac{-3}{2}}{4(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{5}{2}}}\\=&\frac{3}{128(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{5}{2}}x^{\frac{1}{2}}} - \frac{1}{16(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}x^{\frac{3}{2}}} + \frac{3}{8(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{5}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3}{128(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{5}{2}}x^{\frac{1}{2}}} - \frac{1}{16(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}x^{\frac{3}{2}}} + \frac{3}{8(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{5}{2}}}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(\frac{\frac{-1}{4}*2(x)^{\frac{1}{2}}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + 0)}{(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{7}{2}}})}{128x^{\frac{1}{2}}} + \frac{3*\frac{-1}{2}}{128(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{5}{2}}x^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(\frac{\frac{-1}{4}*2(x)^{\frac{1}{2}}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + 0)}{(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{5}{2}}})}{16x^{\frac{3}{2}}} - \frac{\frac{-3}{2}}{16(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}x^{\frac{5}{2}}} + \frac{3(\frac{\frac{-1}{2}(\frac{\frac{-1}{4}*2(x)^{\frac{1}{2}}*\frac{1}{2}}{(x)^{\frac{1}{2}}} + 0)}{(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}})}{8x^{\frac{5}{2}}} + \frac{3*\frac{-5}{2}}{8(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{7}{2}}}\\=&\frac{15}{1024(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{7}{2}}x^{\frac{1}{2}}} - \frac{9}{256(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{5}{2}}x^{\frac{3}{2}}} + \frac{9}{64(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{3}{2}}x^{\frac{5}{2}}} - \frac{15}{16(\frac{-1}{4}sqrt(x)^{2} + 1)^{\frac{1}{2}}x^{\frac{7}{2}}}\\ \end{split}\end{equation} \]

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