There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{cot(x)}{tan(x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{cot(x)}{tan(x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{cot(x)}{tan(x)}\right)}{dx}\\=&\frac{-sec^{2}(x)(1)cot(x)}{tan^{2}(x)} + \frac{-csc^{2}(x)}{tan(x)}\\=&\frac{-cot(x)sec^{2}(x)}{tan^{2}(x)} - \frac{csc^{2}(x)}{tan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-cot(x)sec^{2}(x)}{tan^{2}(x)} - \frac{csc^{2}(x)}{tan(x)}\right)}{dx}\\=&\frac{--2sec^{2}(x)(1)cot(x)sec^{2}(x)}{tan^{3}(x)} - \frac{-csc^{2}(x)sec^{2}(x)}{tan^{2}(x)} - \frac{cot(x)*2sec^{2}(x)tan(x)}{tan^{2}(x)} - \frac{-sec^{2}(x)(1)csc^{2}(x)}{tan^{2}(x)} - \frac{-2csc^{2}(x)cot(x)}{tan(x)}\\=&\frac{2cot(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{2cot(x)sec^{2}(x)}{tan(x)} + \frac{2cot(x)csc^{2}(x)}{tan(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2cot(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{2cot(x)sec^{2}(x)}{tan(x)} + \frac{2cot(x)csc^{2}(x)}{tan(x)}\right)}{dx}\\=&\frac{2*-3sec^{2}(x)(1)cot(x)sec^{4}(x)}{tan^{4}(x)} + \frac{2*-csc^{2}(x)sec^{4}(x)}{tan^{3}(x)} + \frac{2cot(x)*4sec^{4}(x)tan(x)}{tan^{3}(x)} + \frac{2*-2sec^{2}(x)(1)sec^{2}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{2*2sec^{2}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} + \frac{2sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} - \frac{2*-sec^{2}(x)(1)cot(x)sec^{2}(x)}{tan^{2}(x)} - \frac{2*-csc^{2}(x)sec^{2}(x)}{tan(x)} - \frac{2cot(x)*2sec^{2}(x)tan(x)}{tan(x)} + \frac{2*-sec^{2}(x)(1)cot(x)csc^{2}(x)}{tan^{2}(x)} + \frac{2*-csc^{2}(x)csc^{2}(x)}{tan(x)} + \frac{2cot(x)*-2csc^{2}(x)cot(x)}{tan(x)}\\=& - \frac{6cot(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{10cot(x)sec^{4}(x)}{tan^{2}(x)} + \frac{6sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{6cot(x)sec^{6}(x)}{tan^{4}(x)} - 4cot(x)sec^{2}(x) - \frac{2csc^{4}(x)}{tan(x)} - \frac{4cot^{2}(x)csc^{2}(x)}{tan(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{6cot(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{10cot(x)sec^{4}(x)}{tan^{2}(x)} + \frac{6sec^{2}(x)csc^{2}(x)}{tan(x)} - \frac{6cot(x)sec^{6}(x)}{tan^{4}(x)} - 4cot(x)sec^{2}(x) - \frac{2csc^{4}(x)}{tan(x)} - \frac{4cot^{2}(x)csc^{2}(x)}{tan(x)}\right)}{dx}\\=& - \frac{6*-2sec^{2}(x)(1)cot(x)sec^{2}(x)csc^{2}(x)}{tan^{3}(x)} - \frac{6*-csc^{2}(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6cot(x)*2sec^{2}(x)tan(x)csc^{2}(x)}{tan^{2}(x)} - \frac{6cot(x)sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan^{2}(x)} - \frac{6*-3sec^{2}(x)(1)sec^{4}(x)csc^{2}(x)}{tan^{4}(x)} - \frac{6*4sec^{4}(x)tan(x)csc^{2}(x)}{tan^{3}(x)} - \frac{6sec^{4}(x)*-2csc^{2}(x)cot(x)}{tan^{3}(x)} + \frac{10*-2sec^{2}(x)(1)cot(x)sec^{4}(x)}{tan^{3}(x)} + \frac{10*-csc^{2}(x)sec^{4}(x)}{tan^{2}(x)} + \frac{10cot(x)*4sec^{4}(x)tan(x)}{tan^{2}(x)} + \frac{6*-sec^{2}(x)(1)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{6*2sec^{2}(x)tan(x)csc^{2}(x)}{tan(x)} + \frac{6sec^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)} - \frac{6*-4sec^{2}(x)(1)cot(x)sec^{6}(x)}{tan^{5}(x)} - \frac{6*-csc^{2}(x)sec^{6}(x)}{tan^{4}(x)} - \frac{6cot(x)*6sec^{6}(x)tan(x)}{tan^{4}(x)} - 4*-csc^{2}(x)sec^{2}(x) - 4cot(x)*2sec^{2}(x)tan(x) - \frac{2*-sec^{2}(x)(1)csc^{4}(x)}{tan^{2}(x)} - \frac{2*-4csc^{4}(x)cot(x)}{tan(x)} - \frac{4*-sec^{2}(x)(1)cot^{2}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{4*-2cot(x)csc^{2}(x)csc^{2}(x)}{tan(x)} - \frac{4cot^{2}(x)*-2csc^{2}(x)cot(x)}{tan(x)}\\=&\frac{24cot(x)sec^{4}(x)csc^{2}(x)}{tan^{3}(x)} + \frac{8sec^{2}(x)csc^{4}(x)}{tan^{2}(x)} - \frac{24cot(x)sec^{2}(x)csc^{2}(x)}{tan(x)} + \frac{16cot^{2}(x)sec^{2}(x)csc^{2}(x)}{tan^{2}(x)} + \frac{24sec^{6}(x)csc^{2}(x)}{tan^{4}(x)} - \frac{40sec^{4}(x)csc^{2}(x)}{tan^{2}(x)} - \frac{56cot(x)sec^{6}(x)}{tan^{3}(x)} + \frac{40cot(x)sec^{4}(x)}{tan(x)} + 16sec^{2}(x)csc^{2}(x) + \frac{24cot(x)sec^{8}(x)}{tan^{5}(x)} + \frac{16cot(x)csc^{4}(x)}{tan(x)} - 8tan(x)cot(x)sec^{2}(x) + \frac{8cot^{3}(x)csc^{2}(x)}{tan(x)}\\ \end{split}\end{equation} \]

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