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

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