There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ arcsin(sin(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arcsin(sin(x))\right)}{dx}\\=&(\frac{(cos(x))}{((1 - (sin(x))^{2})^{\frac{1}{2}})})\\=&\frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})cos(x) + \frac{-sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})sin(x)cos^{2}(x) + \frac{cos(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin(x)*-2cos(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})sin(x) - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{3sin^{2}(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3sin^{2}(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{cos(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{7}{2}}})sin^{2}(x)cos^{3}(x) + \frac{3*2sin(x)cos(x)cos^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + \frac{3sin^{2}(x)*-3cos^{2}(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})cos^{3}(x) + \frac{-3cos^{2}(x)sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - 3(\frac{\frac{-3}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}})sin^{2}(x)cos(x) - \frac{3*2sin(x)cos(x)cos(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - \frac{3sin^{2}(x)*-sin(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} - (\frac{\frac{-1}{2}(-2sin(x)cos(x) + 0)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}})cos(x) - \frac{-sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\=&\frac{15sin^{3}(x)cos^{4}(x)}{(-sin^{2}(x) + 1)^{\frac{7}{2}}} + \frac{9sin(x)cos^{4}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} - \frac{18sin^{3}(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{5}{2}}} - \frac{10sin(x)cos^{2}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{3sin^{3}(x)}{(-sin^{2}(x) + 1)^{\frac{3}{2}}} + \frac{sin(x)}{(-sin^{2}(x) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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