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

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