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

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