There are 1 questions in this calculation: for each question, the 2 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ second\ derivative\ of\ function\ \frac{({x}^{2} - 8x + 12)}{({x}^{2} - 8x + 16)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{2}}{(x^{2} - 8x + 16)} - \frac{8x}{(x^{2} - 8x + 16)} + \frac{12}{(x^{2} - 8x + 16)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{2}}{(x^{2} - 8x + 16)} - \frac{8x}{(x^{2} - 8x + 16)} + \frac{12}{(x^{2} - 8x + 16)}\right)}{dx}\\=&(\frac{-(2x - 8 + 0)}{(x^{2} - 8x + 16)^{2}})x^{2} + \frac{2x}{(x^{2} - 8x + 16)} - 8(\frac{-(2x - 8 + 0)}{(x^{2} - 8x + 16)^{2}})x - \frac{8}{(x^{2} - 8x + 16)} + 12(\frac{-(2x - 8 + 0)}{(x^{2} - 8x + 16)^{2}})\\=&\frac{-2x^{3}}{(x^{2} - 8x + 16)^{2}} + \frac{24x^{2}}{(x^{2} - 8x + 16)^{2}} + \frac{2x}{(x^{2} - 8x + 16)} - \frac{88x}{(x^{2} - 8x + 16)^{2}} - \frac{8}{(x^{2} - 8x + 16)} + \frac{96}{(x^{2} - 8x + 16)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x^{3}}{(x^{2} - 8x + 16)^{2}} + \frac{24x^{2}}{(x^{2} - 8x + 16)^{2}} + \frac{2x}{(x^{2} - 8x + 16)} - \frac{88x}{(x^{2} - 8x + 16)^{2}} - \frac{8}{(x^{2} - 8x + 16)} + \frac{96}{(x^{2} - 8x + 16)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x - 8 + 0)}{(x^{2} - 8x + 16)^{3}})x^{3} - \frac{2*3x^{2}}{(x^{2} - 8x + 16)^{2}} + 24(\frac{-2(2x - 8 + 0)}{(x^{2} - 8x + 16)^{3}})x^{2} + \frac{24*2x}{(x^{2} - 8x + 16)^{2}} + 2(\frac{-(2x - 8 + 0)}{(x^{2} - 8x + 16)^{2}})x + \frac{2}{(x^{2} - 8x + 16)} - 88(\frac{-2(2x - 8 + 0)}{(x^{2} - 8x + 16)^{3}})x - \frac{88}{(x^{2} - 8x + 16)^{2}} - 8(\frac{-(2x - 8 + 0)}{(x^{2} - 8x + 16)^{2}}) + 96(\frac{-2(2x - 8 + 0)}{(x^{2} - 8x + 16)^{3}})\\=&\frac{8x^{4}}{(x^{2} - 8x + 16)^{3}} - \frac{128x^{3}}{(x^{2} - 8x + 16)^{3}} - \frac{10x^{2}}{(x^{2} - 8x + 16)^{2}} + \frac{736x^{2}}{(x^{2} - 8x + 16)^{3}} + \frac{80x}{(x^{2} - 8x + 16)^{2}} - \frac{1792x}{(x^{2} - 8x + 16)^{3}} - \frac{152}{(x^{2} - 8x + 16)^{2}} + \frac{2}{(x^{2} - 8x + 16)} + \frac{1536}{(x^{2} - 8x + 16)^{3}}\\ \end{split}\end{equation} \]

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