There are 1 questions in this calculation: for each question, the 3 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ third\ derivative\ of\ function\ \frac{tan(x)}{(1 + {x}^{2})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{tan(x)}{(x^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{tan(x)}{(x^{2} + 1)}\right)}{dx}\\=&(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})tan(x) + \frac{sec^{2}(x)(1)}{(x^{2} + 1)}\\=&\frac{-2xtan(x)}{(x^{2} + 1)^{2}} + \frac{sec^{2}(x)}{(x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2xtan(x)}{(x^{2} + 1)^{2}} + \frac{sec^{2}(x)}{(x^{2} + 1)}\right)}{dx}\\=&-2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xtan(x) - \frac{2tan(x)}{(x^{2} + 1)^{2}} - \frac{2xsec^{2}(x)(1)}{(x^{2} + 1)^{2}} + (\frac{-(2x + 0)}{(x^{2} + 1)^{2}})sec^{2}(x) + \frac{2sec^{2}(x)tan(x)}{(x^{2} + 1)}\\=&\frac{8x^{2}tan(x)}{(x^{2} + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(x^{2} + 1)} - \frac{4xsec^{2}(x)}{(x^{2} + 1)^{2}} - \frac{2tan(x)}{(x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{2}tan(x)}{(x^{2} + 1)^{3}} + \frac{2tan(x)sec^{2}(x)}{(x^{2} + 1)} - \frac{4xsec^{2}(x)}{(x^{2} + 1)^{2}} - \frac{2tan(x)}{(x^{2} + 1)^{2}}\right)}{dx}\\=&8(\frac{-3(2x + 0)}{(x^{2} + 1)^{4}})x^{2}tan(x) + \frac{8*2xtan(x)}{(x^{2} + 1)^{3}} + \frac{8x^{2}sec^{2}(x)(1)}{(x^{2} + 1)^{3}} + 2(\frac{-(2x + 0)}{(x^{2} + 1)^{2}})tan(x)sec^{2}(x) + \frac{2sec^{2}(x)(1)sec^{2}(x)}{(x^{2} + 1)} + \frac{2tan(x)*2sec^{2}(x)tan(x)}{(x^{2} + 1)} - 4(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})xsec^{2}(x) - \frac{4sec^{2}(x)}{(x^{2} + 1)^{2}} - \frac{4x*2sec^{2}(x)tan(x)}{(x^{2} + 1)^{2}} - 2(\frac{-2(2x + 0)}{(x^{2} + 1)^{3}})tan(x) - \frac{2sec^{2}(x)(1)}{(x^{2} + 1)^{2}}\\=&\frac{-12xtan(x)sec^{2}(x)}{(x^{2} + 1)^{2}} + \frac{24xtan(x)}{(x^{2} + 1)^{3}} + \frac{24x^{2}sec^{2}(x)}{(x^{2} + 1)^{3}} - \frac{48x^{3}tan(x)}{(x^{2} + 1)^{4}} + \frac{2sec^{4}(x)}{(x^{2} + 1)} + \frac{4tan^{2}(x)sec^{2}(x)}{(x^{2} + 1)} - \frac{6sec^{2}(x)}{(x^{2} + 1)^{2}}\\ \end{split}\end{equation} \]

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