There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{arcsin(x)}{sin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{arcsin(x)}{sin(x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{arcsin(x)}{sin(x)}\right)}{dx}\\=&\frac{-cos(x)arcsin(x)}{sin^{2}(x)} + \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{sin(x)}\\=&\frac{-cos(x)arcsin(x)}{sin^{2}(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}sin(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-cos(x)arcsin(x)}{sin^{2}(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}sin(x)}\right)}{dx}\\=&\frac{--2cos(x)cos(x)arcsin(x)}{sin^{3}(x)} - \frac{-sin(x)arcsin(x)}{sin^{2}(x)} - \frac{cos(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{sin^{2}(x)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{sin(x)} + \frac{-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)}\\=&\frac{2cos^{2}(x)arcsin(x)}{sin^{3}(x)} + \frac{arcsin(x)}{sin(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)} + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}sin(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2cos^{2}(x)arcsin(x)}{sin^{3}(x)} + \frac{arcsin(x)}{sin(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)} + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}sin(x)} - \frac{cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)}\right)}{dx}\\=&\frac{2*-3cos(x)cos^{2}(x)arcsin(x)}{sin^{4}(x)} + \frac{2*-2cos(x)sin(x)arcsin(x)}{sin^{3}(x)} + \frac{2cos^{2}(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{sin^{3}(x)} + \frac{-cos(x)arcsin(x)}{sin^{2}(x)} + \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{sin(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{sin^{2}(x)} - \frac{-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{3}(x)} - \frac{-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{sin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}sin(x)} + \frac{x*-cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{2}(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(x)}{sin^{2}(x)} - \frac{-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{3}(x)} - \frac{-sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)}\\=&\frac{-6cos^{3}(x)arcsin(x)}{sin^{4}(x)} - \frac{5cos(x)arcsin(x)}{sin^{2}(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{3}(x)} + \frac{4cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{3}(x)} - \frac{3xcos(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{2}(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}sin(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}sin(x)} + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}sin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}sin(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6cos^{3}(x)arcsin(x)}{sin^{4}(x)} - \frac{5cos(x)arcsin(x)}{sin^{2}(x)} + \frac{2cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{3}(x)} + \frac{4cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{3}(x)} - \frac{3xcos(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{2}(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}sin(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}sin(x)} + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}sin(x)} + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}sin(x)}\right)}{dx}\\=&\frac{-6*-4cos(x)cos^{3}(x)arcsin(x)}{sin^{5}(x)} - \frac{6*-3cos^{2}(x)sin(x)arcsin(x)}{sin^{4}(x)} - \frac{6cos^{3}(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{sin^{4}(x)} - \frac{5*-2cos(x)cos(x)arcsin(x)}{sin^{3}(x)} - \frac{5*-sin(x)arcsin(x)}{sin^{2}(x)} - \frac{5cos(x)(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{sin^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{sin^{3}(x)} + \frac{2*-3cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{4}(x)} + \frac{2*-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{3}(x)} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos^{2}(x)}{sin^{3}(x)} + \frac{4*-3cos(x)cos^{2}(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{4}(x)} + \frac{4*-2cos(x)sin(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{3}(x)} - \frac{3(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcos(x)}{sin^{2}(x)} - \frac{3cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{2}(x)} - \frac{3x*-2cos(x)cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{3}(x)} - \frac{3x*-sin(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{2}(x)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{sin(x)} + \frac{-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{sin(x)} + \frac{2*-cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)} + \frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}}{sin(x)} + \frac{3*2x}{(-x^{2} + 1)^{\frac{5}{2}}sin(x)} + \frac{3x^{2}*-cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}sin^{2}(x)} + \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})}{sin(x)} + \frac{-cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{2}(x)}\\=&\frac{24cos^{4}(x)arcsin(x)}{sin^{5}(x)} + \frac{28cos^{2}(x)arcsin(x)}{sin^{3}(x)} - \frac{6cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{4}(x)} + \frac{5arcsin(x)}{sin(x)} - \frac{5cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)} + \frac{12xcos^{2}(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{3}(x)} - \frac{18cos^{3}(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{4}(x)} - \frac{15cos(x)}{(-x^{2} + 1)^{\frac{1}{2}}sin^{2}(x)} - \frac{12x^{2}cos(x)}{(-x^{2} + 1)^{\frac{5}{2}}sin^{2}(x)} - \frac{4cos(x)}{(-x^{2} + 1)^{\frac{3}{2}}sin^{2}(x)} + \frac{6x}{(-x^{2} + 1)^{\frac{3}{2}}sin(x)} + \frac{15x^{3}}{(-x^{2} + 1)^{\frac{7}{2}}sin(x)} + \frac{9x}{(-x^{2} + 1)^{\frac{5}{2}}sin(x)}\\ \end{split}\end{equation} \]

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