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

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