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

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