There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ cos(arcsin(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( cos(arcsin(x))\right)}{dx}\\=&-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{-sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(arcsin(x)) - \frac{cos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xsin(arcsin(x)) - \frac{sin(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xcos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-3x^{2}sin(arcsin(x))}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{sin(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{2xcos(arcsin(x))}{(-x^{2} + 1)^{2}} + \frac{sin(arcsin(x))}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{2}sin(arcsin(x))}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{sin(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{2xcos(arcsin(x))}{(-x^{2} + 1)^{2}} + \frac{sin(arcsin(x))}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&-3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}sin(arcsin(x)) - \frac{3*2xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}cos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} - (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})sin(arcsin(x)) - \frac{cos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{x*-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xcos(arcsin(x)) - \frac{2cos(arcsin(x))}{(-x^{2} + 1)^{2}} - \frac{2x*-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(arcsin(x))}{(-x^{2} + 1)} + \frac{cos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-15x^{3}sin(arcsin(x))}{(-x^{2} + 1)^{\frac{7}{2}}} - \frac{6xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{12x^{2}cos(arcsin(x))}{(-x^{2} + 1)^{3}} - \frac{3cos(arcsin(x))}{(-x^{2} + 1)^{2}} + \frac{xsin(arcsin(x))}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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