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

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