There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ln(tan(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(tan(x))\right)}{dx}\\=&\frac{sec^{2}(x)(1)}{(tan(x))}\\=&\frac{sec^{2}(x)}{tan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{sec^{2}(x)}{tan(x)}\right)}{dx}\\=&\frac{-sec^{2}(x)(1)sec^{2}(x)}{tan^{2}(x)} + \frac{2sec^{2}(x)tan(x)}{tan(x)}\\=&\frac{-sec^{4}(x)}{tan^{2}(x)} + 2sec^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-sec^{4}(x)}{tan^{2}(x)} + 2sec^{2}(x)\right)}{dx}\\=&\frac{--2sec^{2}(x)(1)sec^{4}(x)}{tan^{3}(x)} - \frac{4sec^{4}(x)tan(x)}{tan^{2}(x)} + 2*2sec^{2}(x)tan(x)\\=&\frac{2sec^{6}(x)}{tan^{3}(x)} - \frac{4sec^{4}(x)}{tan(x)} + 4tan(x)sec^{2}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2sec^{6}(x)}{tan^{3}(x)} - \frac{4sec^{4}(x)}{tan(x)} + 4tan(x)sec^{2}(x)\right)}{dx}\\=&\frac{2*-3sec^{2}(x)(1)sec^{6}(x)}{tan^{4}(x)} + \frac{2*6sec^{6}(x)tan(x)}{tan^{3}(x)} - \frac{4*-sec^{2}(x)(1)sec^{4}(x)}{tan^{2}(x)} - \frac{4*4sec^{4}(x)tan(x)}{tan(x)} + 4sec^{2}(x)(1)sec^{2}(x) + 4tan(x)*2sec^{2}(x)tan(x)\\=&\frac{-6sec^{8}(x)}{tan^{4}(x)} + \frac{16sec^{6}(x)}{tan^{2}(x)} - 12sec^{4}(x) + 8tan^{2}(x)sec^{2}(x)\\ \end{split}\end{equation} \]

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