There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(lg(x))arcsin(x)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(lg(x))arcsin(x)\right)}{dx}\\=&\frac{arcsin(x)}{ln{10}(lg(x))ln{10}(x)} + lg(lg(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{arcsin(x)}{xln^{2}{10}lg(x)} + \frac{lg(lg(x))}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{arcsin(x)}{xln^{2}{10}lg(x)} + \frac{lg(lg(x))}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{-arcsin(x)}{x^{2}ln^{2}{10}lg(x)} + \frac{-2*0arcsin(x)}{xln^{3}{10}lg(x)} + \frac{-arcsin(x)}{xln^{2}{10}lg^{2}(x)ln{10}(x)} + \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{xln^{2}{10}lg(x)} + (\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})lg(lg(x)) + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}ln{10}(lg(x))ln{10}(x)}\\=&\frac{-arcsin(x)}{x^{2}ln^{2}{10}lg(x)} - \frac{arcsin(x)}{x^{2}ln^{3}{10}lg^{2}(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}lg(x)} + \frac{xlg(lg(x))}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}lg(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-arcsin(x)}{x^{2}ln^{2}{10}lg(x)} - \frac{arcsin(x)}{x^{2}ln^{3}{10}lg^{2}(x)} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}lg(x)} + \frac{xlg(lg(x))}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}lg(x)}\right)}{dx}\\=&\frac{--2arcsin(x)}{x^{3}ln^{2}{10}lg(x)} - \frac{-2*0arcsin(x)}{x^{2}ln^{3}{10}lg(x)} - \frac{-arcsin(x)}{x^{2}ln^{2}{10}lg^{2}(x)ln{10}(x)} - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{2}ln^{2}{10}lg(x)} - \frac{-2arcsin(x)}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{-3*0arcsin(x)}{x^{2}ln^{4}{10}lg^{2}(x)} - \frac{-2arcsin(x)}{x^{2}ln^{3}{10}lg^{3}(x)ln{10}(x)} - \frac{(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{2}ln^{3}{10}lg^{2}(x)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{xln^{2}{10}lg(x)} + \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}lg(x)} + \frac{-2*0}{(-x^{2} + 1)^{\frac{1}{2}}xln^{3}{10}lg(x)} + \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}lg^{2}(x)ln{10}(x)} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xlg(lg(x)) + \frac{lg(lg(x))}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}(lg(x))ln{10}(x)} + \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{xln^{2}{10}lg(x)} + \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}lg(x)} + \frac{-2*0}{(-x^{2} + 1)^{\frac{1}{2}}xln^{3}{10}lg(x)} + \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}xln^{2}{10}lg^{2}(x)ln{10}(x)}\\=&\frac{2arcsin(x)}{x^{3}ln^{2}{10}lg(x)} + \frac{3arcsin(x)}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}lg(x)} + \frac{2arcsin(x)}{x^{3}ln^{4}{10}lg^{3}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{3}{10}lg^{2}(x)} + \frac{3}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}{10}lg(x)} - \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}lg(x)} - \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{3}{10}lg^{2}(x)} + \frac{3x^{2}lg(lg(x))}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{lg(lg(x))}{(-x^{2} + 1)^{\frac{3}{2}}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2arcsin(x)}{x^{3}ln^{2}{10}lg(x)} + \frac{3arcsin(x)}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}lg(x)} + \frac{2arcsin(x)}{x^{3}ln^{4}{10}lg^{3}(x)} - \frac{1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{3}{10}lg^{2}(x)} + \frac{3}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}{10}lg(x)} - \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}lg(x)} - \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{3}{10}lg^{2}(x)} + \frac{3x^{2}lg(lg(x))}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{lg(lg(x))}{(-x^{2} + 1)^{\frac{3}{2}}}\right)}{dx}\\=&\frac{2*-3arcsin(x)}{x^{4}ln^{2}{10}lg(x)} + \frac{2*-2*0arcsin(x)}{x^{3}ln^{3}{10}lg(x)} + \frac{2*-arcsin(x)}{x^{3}ln^{2}{10}lg^{2}(x)ln{10}(x)} + \frac{2(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{3}ln^{2}{10}lg(x)} + \frac{3*-3arcsin(x)}{x^{4}ln^{3}{10}lg^{2}(x)} + \frac{3*-3*0arcsin(x)}{x^{3}ln^{4}{10}lg^{2}(x)} + \frac{3*-2arcsin(x)}{x^{3}ln^{3}{10}lg^{3}(x)ln{10}(x)} + \frac{3(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{3}ln^{3}{10}lg^{2}(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln^{2}{10}lg(x)} - \frac{-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{2}{10}lg(x)} - \frac{-2*0}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{3}{10}lg(x)} - \frac{-1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}lg^{2}(x)ln{10}(x)} + \frac{2*-3arcsin(x)}{x^{4}ln^{4}{10}lg^{3}(x)} + \frac{2*-4*0arcsin(x)}{x^{3}ln^{5}{10}lg^{3}(x)} + \frac{2*-3arcsin(x)}{x^{3}ln^{4}{10}lg^{4}(x)ln{10}(x)} + \frac{2(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{x^{3}ln^{4}{10}lg^{3}(x)} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln^{3}{10}lg^{2}(x)} - \frac{-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{3}{10}lg^{2}(x)} - \frac{-3*0}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{4}{10}lg^{2}(x)} - \frac{-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{3}{10}lg^{3}(x)ln{10}(x)} + \frac{3(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})}{ln^{2}{10}lg(x)} + \frac{3*-2*0}{(-x^{2} + 1)^{\frac{3}{2}}ln^{3}{10}lg(x)} + \frac{3*-1}{(-x^{2} + 1)^{\frac{3}{2}}ln^{2}{10}lg^{2}(x)ln{10}(x)} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln^{2}{10}lg(x)} - \frac{2*-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{2}{10}lg(x)} - \frac{2*-2*0}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{3}{10}lg(x)} - \frac{2*-1}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{2}{10}lg^{2}(x)ln{10}(x)} - \frac{2(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})}{x^{2}ln^{3}{10}lg^{2}(x)} - \frac{2*-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{3}{10}lg^{2}(x)} - \frac{2*-3*0}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{4}{10}lg^{2}(x)} - \frac{2*-2}{(-x^{2} + 1)^{\frac{1}{2}}x^{2}ln^{3}{10}lg^{3}(x)ln{10}(x)} + 3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}lg(lg(x)) + \frac{3*2xlg(lg(x))}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}}{(-x^{2} + 1)^{\frac{5}{2}}ln{10}(lg(x))ln{10}(x)} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})lg(lg(x)) + \frac{1}{(-x^{2} + 1)^{\frac{3}{2}}ln{10}(lg(x))ln{10}(x)}\\=&\frac{-6arcsin(x)}{x^{4}ln^{2}{10}lg(x)} - \frac{11arcsin(x)}{x^{4}ln^{3}{10}lg^{2}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{2}{10}lg(x)} - \frac{12arcsin(x)}{x^{4}ln^{4}{10}lg^{3}(x)} + \frac{3}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{3}{10}lg^{2}(x)} - \frac{2}{(-x^{2} + 1)^{\frac{3}{2}}xln^{2}{10}lg(x)} + \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{2}{10}lg(x)} + \frac{9}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{3}{10}lg^{2}(x)} - \frac{6arcsin(x)}{x^{4}ln^{5}{10}lg^{4}(x)} + \frac{2}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{4}{10}lg^{3}(x)} - \frac{6}{(-x^{2} + 1)^{\frac{3}{2}}xln^{3}{10}lg^{2}(x)} + \frac{6}{(-x^{2} + 1)^{\frac{1}{2}}x^{3}ln^{4}{10}lg^{3}(x)} + \frac{12x}{(-x^{2} + 1)^{\frac{5}{2}}ln^{2}{10}lg(x)} + \frac{15x^{3}lg(lg(x))}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{9xlg(lg(x))}{(-x^{2} + 1)^{\frac{5}{2}}}\\ \end{split}\end{equation} \]

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