There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ arctan(arctan(x))\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( arctan(arctan(x))\right)}{dx}\\=&(\frac{((\frac{(1)}{(1 + (x)^{2})}))}{(1 + (arctan(x))^{2})})\\=&\frac{1}{(x^{2} + 1)(arctan^{2}(x) + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{(x^{2} + 1)(arctan^{2}(x) + 1)}\right)}{dx}\\=&\frac{(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})}{(arctan^{2}(x) + 1)} + \frac{(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})}{(x^{2} + 1)}\\=&\frac{-2x}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\right)}{dx}\\=&\frac{-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})x}{(arctan^{2}(x) + 1)} - \frac{2(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})x}{(x^{2} + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)} - \frac{2(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})arctan(x)}{(x^{2} + 1)^{2}} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})arctan(x)}{(arctan^{2}(x) + 1)^{2}} - \frac{2(\frac{(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{2}}\\=&\frac{12xarctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8x^{2}}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{12xarctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8x^{2}}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)^{2}} - \frac{2}{(x^{2} + 1)^{2}(arctan^{2}(x) + 1)}\right)}{dx}\\=&\frac{12(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})xarctan(x)}{(x^{2} + 1)^{3}} + \frac{12(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})xarctan(x)}{(arctan^{2}(x) + 1)^{2}} + \frac{12arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{12x(\frac{(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}}{(arctan^{2}(x) + 1)} + \frac{8(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})x^{2}}{(x^{2} + 1)^{3}} + \frac{8*2x}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)} + \frac{8(\frac{-3((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{4}})arctan^{2}(x)}{(x^{2} + 1)^{3}} + \frac{8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})arctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}} + \frac{8(\frac{2arctan(x)(1)}{(1 + (x)^{2})})}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{3}} - \frac{2(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})}{(arctan^{2}(x) + 1)^{2}} - \frac{2(\frac{-2((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{3}})}{(x^{2} + 1)^{3}} - \frac{2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})}{(arctan^{2}(x) + 1)} - \frac{2(\frac{-((\frac{2arctan(x)(1)}{(1 + (x)^{2})}) + 0)}{(arctan^{2}(x) + 1)^{2}})}{(x^{2} + 1)^{2}}\\=& - \frac{96xarctan^{2}(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{4}} - \frac{88x^{2}arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{4}} + \frac{16arctan(x)}{(arctan^{2}(x) + 1)^{2}(x^{2} + 1)^{3}} + \frac{24x}{(x^{2} + 1)^{4}(arctan^{2}(x) + 1)^{2}} - \frac{48x^{3}}{(x^{2} + 1)^{4}(arctan^{2}(x) + 1)} + \frac{16x}{(arctan^{2}(x) + 1)(x^{2} + 1)^{3}} - \frac{48arctan^{3}(x)}{(arctan^{2}(x) + 1)^{4}(x^{2} + 1)^{4}} + \frac{24arctan(x)}{(arctan^{2}(x) + 1)^{3}(x^{2} + 1)^{4}} + \frac{8x}{(x^{2} + 1)^{3}(arctan^{2}(x) + 1)}\\ \end{split}\end{equation} \]

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