There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(e^{cot(x)})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(e^{cot(x)})\right)}{dx}\\=&\frac{e^{cot(x)}*-csc^{2}(x)}{ln{10}(e^{cot(x)})}\\=&\frac{-csc^{2}(x)}{ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-csc^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{--0csc^{2}(x)}{ln^{2}{10}} - \frac{-2csc^{2}(x)cot(x)}{ln{10}}\\=&\frac{2cot(x)csc^{2}(x)}{ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2cot(x)csc^{2}(x)}{ln{10}}\right)}{dx}\\=&\frac{2*-0cot(x)csc^{2}(x)}{ln^{2}{10}} + \frac{2*-csc^{2}(x)csc^{2}(x)}{ln{10}} + \frac{2cot(x)*-2csc^{2}(x)cot(x)}{ln{10}}\\=& - \frac{2csc^{4}(x)}{ln{10}} - \frac{4cot^{2}(x)csc^{2}(x)}{ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{2csc^{4}(x)}{ln{10}} - \frac{4cot^{2}(x)csc^{2}(x)}{ln{10}}\right)}{dx}\\=& - \frac{2*-0csc^{4}(x)}{ln^{2}{10}} - \frac{2*-4csc^{4}(x)cot(x)}{ln{10}} - \frac{4*-0cot^{2}(x)csc^{2}(x)}{ln^{2}{10}} - \frac{4*-2cot(x)csc^{2}(x)csc^{2}(x)}{ln{10}} - \frac{4cot^{2}(x)*-2csc^{2}(x)cot(x)}{ln{10}}\\=&\frac{16cot(x)csc^{4}(x)}{ln{10}} + \frac{8cot^{3}(x)csc^{2}(x)}{ln{10}}\\ \end{split}\end{equation} \]

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