There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{lg(sin(x))}{ln(cos(x))}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{lg(sin(x))}{ln(cos(x))}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{lg(sin(x))}{ln(cos(x))}\right)}{dx}\\=&\frac{--sin(x)lg(sin(x))}{ln^{2}(cos(x))(cos(x))} + \frac{cos(x)}{ln(cos(x))ln{10}(sin(x))}\\=&\frac{lg(sin(x))sin(x)}{ln^{2}(cos(x))cos(x)} + \frac{cos(x)}{ln{10}ln(cos(x))sin(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{lg(sin(x))sin(x)}{ln^{2}(cos(x))cos(x)} + \frac{cos(x)}{ln{10}ln(cos(x))sin(x)}\right)}{dx}\\=&\frac{-2*-sin(x)lg(sin(x))sin(x)}{ln^{3}(cos(x))(cos(x))cos(x)} + \frac{cos(x)sin(x)}{ln^{2}(cos(x))ln{10}(sin(x))cos(x)} + \frac{lg(sin(x))cos(x)}{ln^{2}(cos(x))cos(x)} + \frac{lg(sin(x))sin(x)sin(x)}{ln^{2}(cos(x))cos^{2}(x)} + \frac{-0cos(x)}{ln^{2}{10}ln(cos(x))sin(x)} + \frac{--sin(x)cos(x)}{ln{10}ln^{2}(cos(x))(cos(x))sin(x)} + \frac{-cos(x)cos(x)}{ln{10}ln(cos(x))sin^{2}(x)} + \frac{-sin(x)}{ln{10}ln(cos(x))sin(x)}\\=&\frac{2lg(sin(x))sin^{2}(x)}{ln^{3}(cos(x))cos^{2}(x)} - \frac{cos^{2}(x)}{ln{10}ln(cos(x))sin^{2}(x)} + \frac{lg(sin(x))sin^{2}(x)}{ln^{2}(cos(x))cos^{2}(x)} + \frac{lg(sin(x))}{ln^{2}(cos(x))} + \frac{2}{ln{10}ln^{2}(cos(x))} - \frac{1}{ln{10}ln(cos(x))}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2lg(sin(x))sin^{2}(x)}{ln^{3}(cos(x))cos^{2}(x)} - \frac{cos^{2}(x)}{ln{10}ln(cos(x))sin^{2}(x)} + \frac{lg(sin(x))sin^{2}(x)}{ln^{2}(cos(x))cos^{2}(x)} + \frac{lg(sin(x))}{ln^{2}(cos(x))} + \frac{2}{ln{10}ln^{2}(cos(x))} - \frac{1}{ln{10}ln(cos(x))}\right)}{dx}\\=&\frac{2*-3*-sin(x)lg(sin(x))sin^{2}(x)}{ln^{4}(cos(x))(cos(x))cos^{2}(x)} + \frac{2cos(x)sin^{2}(x)}{ln^{3}(cos(x))ln{10}(sin(x))cos^{2}(x)} + \frac{2lg(sin(x))*2sin(x)cos(x)}{ln^{3}(cos(x))cos^{2}(x)} + \frac{2lg(sin(x))sin^{2}(x)*2sin(x)}{ln^{3}(cos(x))cos^{3}(x)} - \frac{-0cos^{2}(x)}{ln^{2}{10}ln(cos(x))sin^{2}(x)} - \frac{--sin(x)cos^{2}(x)}{ln{10}ln^{2}(cos(x))(cos(x))sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{ln{10}ln(cos(x))sin^{3}(x)} - \frac{-2cos(x)sin(x)}{ln{10}ln(cos(x))sin^{2}(x)} + \frac{-2*-sin(x)lg(sin(x))sin^{2}(x)}{ln^{3}(cos(x))(cos(x))cos^{2}(x)} + \frac{cos(x)sin^{2}(x)}{ln^{2}(cos(x))ln{10}(sin(x))cos^{2}(x)} + \frac{lg(sin(x))*2sin(x)cos(x)}{ln^{2}(cos(x))cos^{2}(x)} + \frac{lg(sin(x))sin^{2}(x)*2sin(x)}{ln^{2}(cos(x))cos^{3}(x)} + \frac{-2*-sin(x)lg(sin(x))}{ln^{3}(cos(x))(cos(x))} + \frac{cos(x)}{ln^{2}(cos(x))ln{10}(sin(x))} + \frac{2*-0}{ln^{2}{10}ln^{2}(cos(x))} + \frac{2*-2*-sin(x)}{ln{10}ln^{3}(cos(x))(cos(x))} - \frac{-0}{ln^{2}{10}ln(cos(x))} - \frac{--sin(x)}{ln{10}ln^{2}(cos(x))(cos(x))}\\=&\frac{6lg(sin(x))sin^{3}(x)}{ln^{4}(cos(x))cos^{3}(x)} + \frac{6sin(x)}{ln{10}ln^{3}(cos(x))cos(x)} + \frac{6lg(sin(x))sin(x)}{ln^{3}(cos(x))cos(x)} + \frac{6lg(sin(x))sin^{3}(x)}{ln^{3}(cos(x))cos^{3}(x)} + \frac{2cos^{3}(x)}{ln{10}ln(cos(x))sin^{3}(x)} + \frac{2cos(x)}{ln{10}ln(cos(x))sin(x)} + \frac{2lg(sin(x))sin(x)}{ln^{2}(cos(x))cos(x)} + \frac{2lg(sin(x))sin^{3}(x)}{ln^{2}(cos(x))cos^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{6lg(sin(x))sin^{3}(x)}{ln^{4}(cos(x))cos^{3}(x)} + \frac{6sin(x)}{ln{10}ln^{3}(cos(x))cos(x)} + \frac{6lg(sin(x))sin(x)}{ln^{3}(cos(x))cos(x)} + \frac{6lg(sin(x))sin^{3}(x)}{ln^{3}(cos(x))cos^{3}(x)} + \frac{2cos^{3}(x)}{ln{10}ln(cos(x))sin^{3}(x)} + \frac{2cos(x)}{ln{10}ln(cos(x))sin(x)} + \frac{2lg(sin(x))sin(x)}{ln^{2}(cos(x))cos(x)} + \frac{2lg(sin(x))sin^{3}(x)}{ln^{2}(cos(x))cos^{3}(x)}\right)}{dx}\\=&\frac{6*-4*-sin(x)lg(sin(x))sin^{3}(x)}{ln^{5}(cos(x))(cos(x))cos^{3}(x)} + \frac{6cos(x)sin^{3}(x)}{ln^{4}(cos(x))ln{10}(sin(x))cos^{3}(x)} + \frac{6lg(sin(x))*3sin^{2}(x)cos(x)}{ln^{4}(cos(x))cos^{3}(x)} + \frac{6lg(sin(x))sin^{3}(x)*3sin(x)}{ln^{4}(cos(x))cos^{4}(x)} + \frac{6*-0sin(x)}{ln^{2}{10}ln^{3}(cos(x))cos(x)} + \frac{6*-3*-sin(x)sin(x)}{ln{10}ln^{4}(cos(x))(cos(x))cos(x)} + \frac{6cos(x)}{ln{10}ln^{3}(cos(x))cos(x)} + \frac{6sin(x)sin(x)}{ln{10}ln^{3}(cos(x))cos^{2}(x)} + \frac{6*-3*-sin(x)lg(sin(x))sin(x)}{ln^{4}(cos(x))(cos(x))cos(x)} + \frac{6cos(x)sin(x)}{ln^{3}(cos(x))ln{10}(sin(x))cos(x)} + \frac{6lg(sin(x))cos(x)}{ln^{3}(cos(x))cos(x)} + \frac{6lg(sin(x))sin(x)sin(x)}{ln^{3}(cos(x))cos^{2}(x)} + \frac{6*-3*-sin(x)lg(sin(x))sin^{3}(x)}{ln^{4}(cos(x))(cos(x))cos^{3}(x)} + \frac{6cos(x)sin^{3}(x)}{ln^{3}(cos(x))ln{10}(sin(x))cos^{3}(x)} + \frac{6lg(sin(x))*3sin^{2}(x)cos(x)}{ln^{3}(cos(x))cos^{3}(x)} + \frac{6lg(sin(x))sin^{3}(x)*3sin(x)}{ln^{3}(cos(x))cos^{4}(x)} + \frac{2*-0cos^{3}(x)}{ln^{2}{10}ln(cos(x))sin^{3}(x)} + \frac{2*--sin(x)cos^{3}(x)}{ln{10}ln^{2}(cos(x))(cos(x))sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{ln{10}ln(cos(x))sin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{ln{10}ln(cos(x))sin^{3}(x)} + \frac{2*-0cos(x)}{ln^{2}{10}ln(cos(x))sin(x)} + \frac{2*--sin(x)cos(x)}{ln{10}ln^{2}(cos(x))(cos(x))sin(x)} + \frac{2*-cos(x)cos(x)}{ln{10}ln(cos(x))sin^{2}(x)} + \frac{2*-sin(x)}{ln{10}ln(cos(x))sin(x)} + \frac{2*-2*-sin(x)lg(sin(x))sin(x)}{ln^{3}(cos(x))(cos(x))cos(x)} + \frac{2cos(x)sin(x)}{ln^{2}(cos(x))ln{10}(sin(x))cos(x)} + \frac{2lg(sin(x))cos(x)}{ln^{2}(cos(x))cos(x)} + \frac{2lg(sin(x))sin(x)sin(x)}{ln^{2}(cos(x))cos^{2}(x)} + \frac{2*-2*-sin(x)lg(sin(x))sin^{3}(x)}{ln^{3}(cos(x))(cos(x))cos^{3}(x)} + \frac{2cos(x)sin^{3}(x)}{ln^{2}(cos(x))ln{10}(sin(x))cos^{3}(x)} + \frac{2lg(sin(x))*3sin^{2}(x)cos(x)}{ln^{2}(cos(x))cos^{3}(x)} + \frac{2lg(sin(x))sin^{3}(x)*3sin(x)}{ln^{2}(cos(x))cos^{4}(x)}\\=&\frac{24lg(sin(x))sin^{4}(x)}{ln^{5}(cos(x))cos^{4}(x)} + \frac{24sin^{2}(x)}{ln{10}ln^{4}(cos(x))cos^{2}(x)} + \frac{36lg(sin(x))sin^{2}(x)}{ln^{4}(cos(x))cos^{2}(x)} + \frac{36lg(sin(x))sin^{4}(x)}{ln^{4}(cos(x))cos^{4}(x)} + \frac{12sin^{2}(x)}{ln{10}ln^{3}(cos(x))cos^{2}(x)} - \frac{6cos^{4}(x)}{ln{10}ln(cos(x))sin^{4}(x)} + \frac{2cos^{2}(x)}{ln{10}ln^{2}(cos(x))sin^{2}(x)} + \frac{28lg(sin(x))sin^{2}(x)}{ln^{3}(cos(x))cos^{2}(x)} + \frac{22lg(sin(x))sin^{4}(x)}{ln^{3}(cos(x))cos^{4}(x)} + \frac{8lg(sin(x))sin^{2}(x)}{ln^{2}(cos(x))cos^{2}(x)} - \frac{8cos^{2}(x)}{ln{10}ln(cos(x))sin^{2}(x)} + \frac{2sin^{2}(x)}{ln{10}ln^{2}(cos(x))cos^{2}(x)} + \frac{6}{ln^{3}(cos(x))ln{10}} + \frac{4}{ln{10}ln^{2}(cos(x))} - \frac{2}{ln{10}ln(cos(x))} + \frac{6lg(sin(x))sin^{4}(x)}{ln^{2}(cos(x))cos^{4}(x)} + \frac{6lg(sin(x))}{ln^{3}(cos(x))} + \frac{6}{ln{10}ln^{3}(cos(x))} + \frac{2lg(sin(x))}{ln^{2}(cos(x))}\\ \end{split}\end{equation} \]

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