There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ -arcsin(ln(cos(x)))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( -arcsin(ln(cos(x)))\right)}{dx}\\=&-(\frac{(\frac{-sin(x)}{(cos(x))})}{((1 - (ln(cos(x)))^{2})^{\frac{1}{2}})})\\=&\frac{sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos(x)}\right)}{dx}\\=&\frac{(\frac{\frac{-1}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}})sin(x)}{cos(x)} + \frac{cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos(x)} + \frac{sin(x)sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{2}(x)}\\=&\frac{-ln(cos(x))sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{1}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-ln(cos(x))sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{1}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}}\right)}{dx}\\=&\frac{-(\frac{\frac{-3}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}})ln(cos(x))sin^{2}(x)}{cos^{2}(x)} - \frac{-sin(x)sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}(cos(x))cos^{2}(x)} - \frac{ln(cos(x))*2sin(x)cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{2}(x)} - \frac{ln(cos(x))sin^{2}(x)*2sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{3}(x)} + \frac{(\frac{\frac{-1}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}})sin^{2}(x)}{cos^{2}(x)} + \frac{2sin(x)cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{sin^{2}(x)*2sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{3}(x)} + (\frac{\frac{-1}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}})\\=&\frac{3ln^{2}(cos(x))sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}cos^{3}(x)} + \frac{sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{3}(x)} - \frac{3ln(cos(x))sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos(x)} - \frac{3ln(cos(x))sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{3}(x)} + \frac{2sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos(x)} + \frac{2sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{3ln^{2}(cos(x))sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}cos^{3}(x)} + \frac{sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{3}(x)} - \frac{3ln(cos(x))sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos(x)} - \frac{3ln(cos(x))sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{3}(x)} + \frac{2sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos(x)} + \frac{2sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{3}(x)}\right)}{dx}\\=&\frac{3(\frac{\frac{-5}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{7}{2}}})ln^{2}(cos(x))sin^{3}(x)}{cos^{3}(x)} + \frac{3*2ln(cos(x))*-sin(x)sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}(cos(x))cos^{3}(x)} + \frac{3ln^{2}(cos(x))*3sin^{2}(x)cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}cos^{3}(x)} + \frac{3ln^{2}(cos(x))sin^{3}(x)*3sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}cos^{4}(x)} + \frac{(\frac{\frac{-3}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}})sin^{3}(x)}{cos^{3}(x)} + \frac{3sin^{2}(x)cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{3}(x)} + \frac{sin^{3}(x)*3sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{4}(x)} - \frac{3(\frac{\frac{-3}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}})ln(cos(x))sin(x)}{cos(x)} - \frac{3*-sin(x)sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}(cos(x))cos(x)} - \frac{3ln(cos(x))cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos(x)} - \frac{3ln(cos(x))sin(x)sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{2}(x)} - \frac{3(\frac{\frac{-3}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}})ln(cos(x))sin^{3}(x)}{cos^{3}(x)} - \frac{3*-sin(x)sin^{3}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}(cos(x))cos^{3}(x)} - \frac{3ln(cos(x))*3sin^{2}(x)cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{3}(x)} - \frac{3ln(cos(x))sin^{3}(x)*3sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{4}(x)} + \frac{2(\frac{\frac{-1}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}})sin(x)}{cos(x)} + \frac{2cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos(x)} + \frac{2sin(x)sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{2(\frac{\frac{-1}{2}(\frac{-2ln(cos(x))*-sin(x)}{(cos(x))} + 0)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}})sin^{3}(x)}{cos^{3}(x)} + \frac{2*3sin^{2}(x)cos(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{3}(x)} + \frac{2sin^{3}(x)*3sin(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{4}(x)}\\=&\frac{-15ln^{3}(cos(x))sin^{4}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{7}{2}}cos^{4}(x)} - \frac{9ln(cos(x))sin^{4}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}cos^{4}(x)} + \frac{18ln^{2}(cos(x))sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}cos^{2}(x)} + \frac{18ln^{2}(cos(x))sin^{4}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{5}{2}}cos^{4}(x)} + \frac{6sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{2}(x)} + \frac{6sin^{4}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{4}(x)} - \frac{14ln(cos(x))sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{2}(x)} - \frac{11ln(cos(x))sin^{4}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}cos^{4}(x)} - \frac{3ln(cos(x))}{(-ln^{2}(cos(x)) + 1)^{\frac{3}{2}}} + \frac{8sin^{2}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{2}(x)} + \frac{6sin^{4}(x)}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}cos^{4}(x)} + \frac{2}{(-ln^{2}(cos(x)) + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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