There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{1}{tan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{tan(x)}\right)}{dx}\\=&\frac{-sec^{2}(x)(1)}{tan^{2}(x)}\\=&\frac{-sec^{2}(x)}{tan^{2}(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sec^{2}(x)}{tan^{2}(x)}\right)}{dx}\\=&\frac{--2sec^{2}(x)(1)sec^{2}(x)}{tan^{3}(x)} - \frac{2sec^{2}(x)tan(x)}{tan^{2}(x)}\\=&\frac{2sec^{4}(x)}{tan^{3}(x)} - \frac{2sec^{2}(x)}{tan(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sec^{4}(x)}{tan^{3}(x)} - \frac{2sec^{2}(x)}{tan(x)}\right)}{dx}\\=&\frac{2*-3sec^{2}(x)(1)sec^{4}(x)}{tan^{4}(x)} + \frac{2*4sec^{4}(x)tan(x)}{tan^{3}(x)} - \frac{2*-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} - \frac{2*2sec^{2}(x)tan(x)}{tan(x)}\\=&\frac{-6sec^{6}(x)}{tan^{4}(x)} + \frac{10sec^{4}(x)}{tan^{2}(x)} - 4sec^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6sec^{6}(x)}{tan^{4}(x)} + \frac{10sec^{4}(x)}{tan^{2}(x)} - 4sec^{2}(x)\right)}{dx}\\=&\frac{-6*-4sec^{2}(x)(1)sec^{6}(x)}{tan^{5}(x)} - \frac{6*6sec^{6}(x)tan(x)}{tan^{4}(x)} + \frac{10*-2sec^{2}(x)(1)sec^{4}(x)}{tan^{3}(x)} + \frac{10*4sec^{4}(x)tan(x)}{tan^{2}(x)} - 4*2sec^{2}(x)tan(x)\\=&\frac{24sec^{8}(x)}{tan^{5}(x)} - \frac{56sec^{6}(x)}{tan^{3}(x)} + \frac{40sec^{4}(x)}{tan(x)} - 8tan(x)sec^{2}(x)\\ \end{split}\end{equation} \]

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