There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ln(arctan(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( ln(arctan(x))\right)}{dx}\\=&\frac{(\frac{(1)}{(1 + (x)^{2})})}{(arctan(x))}\\=&\frac{1}{(x^{2} + 1)arctan(x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{(x^{2} + 1)arctan(x)}\right)}{dx}\\=&\frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{arctan(x)} + \frac{(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)}\\=&\frac{-2x}{(x^{2} + 1)^{2}arctan(x)} - \frac{1}{(x^{2} + 1)^{2}arctan^{2}(x)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x}{(x^{2} + 1)^{2}arctan(x)} - \frac{1}{(x^{2} + 1)^{2}arctan^{2}(x)}\right)}{dx}\\=&\frac{-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x}{arctan(x)} - \frac{2}{(x^{2} + 1)^{2}arctan(x)} - \frac{2x(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} - \frac{(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})}{arctan^{2}(x)} - \frac{(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}}\\=&\frac{8x^{2}}{(x^{2} + 1)^{3}arctan(x)} - \frac{2}{(x^{2} + 1)^{2}arctan(x)} + \frac{6x}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{2}{(x^{2} + 1)^{3}arctan^{3}(x)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{2}}{(x^{2} + 1)^{3}arctan(x)} - \frac{2}{(x^{2} + 1)^{2}arctan(x)} + \frac{6x}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{2}{(x^{2} + 1)^{3}arctan^{3}(x)}\right)}{dx}\\=&\frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}}{arctan(x)} + \frac{8*2x}{(x^{2} + 1)^{3}arctan(x)} + \frac{8x^{2}(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})}{arctan(x)} - \frac{2(\frac{-(1)}{arctan^{2}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{2}} + \frac{6(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x}{arctan^{2}(x)} + \frac{6}{(x^{2} + 1)^{3}arctan^{2}(x)} + \frac{6x(\frac{-2(1)}{arctan^{3}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}} + \frac{2(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})}{arctan^{3}(x)} + \frac{2(\frac{-3(1)}{arctan^{4}(x)(1 + (x)^{2})})}{(x^{2} + 1)^{3}}\\=&\frac{-48x^{3}}{(x^{2} + 1)^{4}arctan(x)} + \frac{24x}{(x^{2} + 1)^{3}arctan(x)} - \frac{44x^{2}}{(x^{2} + 1)^{4}arctan^{2}(x)} + \frac{8}{(x^{2} + 1)^{3}arctan^{2}(x)} - \frac{24x}{(x^{2} + 1)^{4}arctan^{3}(x)} - \frac{6}{(x^{2} + 1)^{4}arctan^{4}(x)}\\ \end{split}\end{equation} \]

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