There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ \frac{({x}^{2} + 2x)}{(ln(\frac{x}{2}) + x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{2}}{(ln(\frac{1}{2}x) + x)} + \frac{2x}{(ln(\frac{1}{2}x) + x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{2}}{(ln(\frac{1}{2}x) + x)} + \frac{2x}{(ln(\frac{1}{2}x) + x)}\right)}{dx}\\=&(\frac{-(\frac{\frac{1}{2}}{(\frac{1}{2}x)} + 1)}{(ln(\frac{1}{2}x) + x)^{2}})x^{2} + \frac{2x}{(ln(\frac{1}{2}x) + x)} + 2(\frac{-(\frac{\frac{1}{2}}{(\frac{1}{2}x)} + 1)}{(ln(\frac{1}{2}x) + x)^{2}})x + \frac{2}{(ln(\frac{1}{2}x) + x)}\\=&\frac{-3x}{(ln(\frac{1}{2}x) + x)^{2}} + \frac{2x}{(ln(\frac{1}{2}x) + x)} - \frac{x^{2}}{(ln(\frac{1}{2}x) + x)^{2}} - \frac{2}{(ln(\frac{1}{2}x) + x)^{2}} + \frac{2}{(ln(\frac{1}{2}x) + x)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x}{(ln(\frac{1}{2}x) + x)^{2}} + \frac{2x}{(ln(\frac{1}{2}x) + x)} - \frac{x^{2}}{(ln(\frac{1}{2}x) + x)^{2}} - \frac{2}{(ln(\frac{1}{2}x) + x)^{2}} + \frac{2}{(ln(\frac{1}{2}x) + x)}\right)}{dx}\\=&-3(\frac{-2(\frac{\frac{1}{2}}{(\frac{1}{2}x)} + 1)}{(ln(\frac{1}{2}x) + x)^{3}})x - \frac{3}{(ln(\frac{1}{2}x) + x)^{2}} + 2(\frac{-(\frac{\frac{1}{2}}{(\frac{1}{2}x)} + 1)}{(ln(\frac{1}{2}x) + x)^{2}})x + \frac{2}{(ln(\frac{1}{2}x) + x)} - (\frac{-2(\frac{\frac{1}{2}}{(\frac{1}{2}x)} + 1)}{(ln(\frac{1}{2}x) + x)^{3}})x^{2} - \frac{2x}{(ln(\frac{1}{2}x) + x)^{2}} - 2(\frac{-2(\frac{\frac{1}{2}}{(\frac{1}{2}x)} + 1)}{(ln(\frac{1}{2}x) + x)^{3}}) + 2(\frac{-(\frac{\frac{1}{2}}{(\frac{1}{2}x)} + 1)}{(ln(\frac{1}{2}x) + x)^{2}})\\=&\frac{8x}{(ln(\frac{1}{2}x) + x)^{3}} - \frac{4x}{(ln(\frac{1}{2}x) + x)^{2}} + \frac{2x^{2}}{(ln(\frac{1}{2}x) + x)^{3}} + \frac{4}{(ln(\frac{1}{2}x) + x)^{3}x} - \frac{2}{(ln(\frac{1}{2}x) + x)^{2}x} + \frac{2}{(ln(\frac{1}{2}x) + x)} - \frac{7}{(ln(\frac{1}{2}x) + x)^{2}} + \frac{10}{(ln(\frac{1}{2}x) + x)^{3}}\\ \end{split}\end{equation} \]

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