There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ \frac{(1 + {(2x)}^{2})arctan(2x)}{2} + 8{e}^{7}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{2}arctan(2x) + 2x^{2}arctan(2x) + 8e^{7}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{2}arctan(2x) + 2x^{2}arctan(2x) + 8e^{7}\right)}{dx}\\=&\frac{1}{2}(\frac{(2)}{(1 + (2x)^{2})}) + 2*2xarctan(2x) + 2x^{2}(\frac{(2)}{(1 + (2x)^{2})}) + 8*7e^{6}*0\\=&\frac{4x^{2}}{(4x^{2} + 1)} + 4xarctan(2x) + \frac{1}{(4x^{2} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{4x^{2}}{(4x^{2} + 1)} + 4xarctan(2x) + \frac{1}{(4x^{2} + 1)}\right)}{dx}\\=&4(\frac{-(4*2x + 0)}{(4x^{2} + 1)^{2}})x^{2} + \frac{4*2x}{(4x^{2} + 1)} + 4arctan(2x) + 4x(\frac{(2)}{(1 + (2x)^{2})}) + (\frac{-(4*2x + 0)}{(4x^{2} + 1)^{2}})\\=& - \frac{32x^{3}}{(4x^{2} + 1)^{2}} + \frac{16x}{(4x^{2} + 1)} + 4arctan(2x) - \frac{8x}{(4x^{2} + 1)^{2}}\\ \end{split}\end{equation} \]

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