There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {tan(x)}^{2}{\frac{1}{sec(x)}}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{tan^{2}(x)}{sec^{2}(x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{tan^{2}(x)}{sec^{2}(x)}\right)}{dx}\\=&\frac{2tan(x)sec^{2}(x)(1)}{sec^{2}(x)} + \frac{tan^{2}(x)*-2tan(x)}{sec^{2}(x)}\\=&\frac{-2tan^{3}(x)}{sec^{2}(x)} + 2tan(x)\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2tan^{3}(x)}{sec^{2}(x)} + 2tan(x)\right)}{dx}\\=&\frac{-2*3tan^{2}(x)sec^{2}(x)(1)}{sec^{2}(x)} - \frac{2tan^{3}(x)*-2tan(x)}{sec^{2}(x)} + 2sec^{2}(x)(1)\\=&\frac{4tan^{4}(x)}{sec^{2}(x)} - 6tan^{2}(x) + 2sec^{2}(x)\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{4tan^{4}(x)}{sec^{2}(x)} - 6tan^{2}(x) + 2sec^{2}(x)\right)}{dx}\\=&\frac{4*4tan^{3}(x)sec^{2}(x)(1)}{sec^{2}(x)} + \frac{4tan^{4}(x)*-2tan(x)}{sec^{2}(x)} - 6*2tan(x)sec^{2}(x)(1) + 2*2sec^{2}(x)tan(x)\\=& - \frac{8tan^{5}(x)}{sec^{2}(x)} - 8tan(x)sec^{2}(x) + 16tan^{3}(x)\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( - \frac{8tan^{5}(x)}{sec^{2}(x)} - 8tan(x)sec^{2}(x) + 16tan^{3}(x)\right)}{dx}\\=& - \frac{8*5tan^{4}(x)sec^{2}(x)(1)}{sec^{2}(x)} - \frac{8tan^{5}(x)*-2tan(x)}{sec^{2}(x)} - 8sec^{2}(x)(1)sec^{2}(x) - 8tan(x)*2sec^{2}(x)tan(x) + 16*3tan^{2}(x)sec^{2}(x)(1)\\=&\frac{16tan^{6}(x)}{sec^{2}(x)} + 32tan^{2}(x)sec^{2}(x) - 8sec^{4}(x) - 40tan^{4}(x)\\ \end{split}\end{equation} \]

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