There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{x}{arcsin(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{arcsin(x)}\right)}{dx}\\=&\frac{1}{arcsin(x)} + x(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{1}{arcsin(x)} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{arcsin(x)} - \frac{x}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)}\right)}{dx}\\=&(\frac{-(1)}{arcsin^{2}(x)((1 - (x)^{2})^{\frac{1}{2}})}) - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)} - \frac{x(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)} - \frac{x^{2}}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)} + \frac{2x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)} - \frac{x^{2}}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{2}(x)} + \frac{2x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)}\right)}{dx}\\=&\frac{-(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{arcsin^{2}(x)} - \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x^{2}}{arcsin^{2}(x)} - \frac{2x}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{x^{2}(\frac{-2(1)}{arcsin^{3}(x)((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}}})}{arcsin^{2}(x)} - \frac{(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2x(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-4x}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} + \frac{4}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} - \frac{3x^{3}}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} + \frac{2x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{4x^{2}}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)arcsin^{3}(x)} - \frac{6x}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} + \frac{4}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} - \frac{3x^{3}}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} + \frac{2x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{4x^{2}}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{2}{(-x^{2} + 1)arcsin^{3}(x)} - \frac{6x}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)}\right)}{dx}\\=&\frac{-4(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x}{arcsin^{2}(x)} - \frac{4}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} - \frac{4x(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{4(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{4(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{3}}{arcsin^{2}(x)} - \frac{3*3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} - \frac{3x^{3}(\frac{-2(1)}{arcsin^{3}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x^{2}}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{2(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{3}(x)} + \frac{2*2x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{2x^{2}(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} + \frac{4(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})x^{2}}{arcsin^{3}(x)} + \frac{4*2x}{(-x^{2} + 1)^{2}arcsin^{3}(x)} + \frac{4x^{2}(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} + \frac{2(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})}{arcsin^{3}(x)} + \frac{2(\frac{-3(1)}{arcsin^{4}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)} - \frac{6(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})x}{(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)} - \frac{6(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})x}{(-x^{2} + 1)arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)} - \frac{6x(\frac{-4(1)}{arcsin^{5}(x)((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{-21x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{2}(x)} - \frac{4}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{2}(x)} + \frac{8x}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}arcsin^{3}(x)} + \frac{24x}{(-x^{2} + 1)^{2}arcsin^{3}(x)} - \frac{12}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)} - \frac{15x^{4}}{(-x^{2} + 1)^{\frac{7}{2}}arcsin^{2}(x)} + \frac{6x^{3}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}arcsin^{3}(x)} + \frac{24x^{3}}{(-x^{2} + 1)^{3}arcsin^{3}(x)} - \frac{6x^{2}}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{4}(x)} - \frac{12x^{2}}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)arcsin^{4}(x)} - \frac{18x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}arcsin^{4}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{3}{2}}arcsin^{4}(x)} + \frac{24x}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}arcsin^{5}(x)}\\ \end{split}\end{equation} \]

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