There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{1}{2}ln(xx + x + 7)\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{2}ln(x^{2} + x + 7)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{2}ln(x^{2} + x + 7)\right)}{dx}\\=&\frac{\frac{1}{2}(2x + 1 + 0)}{(x^{2} + x + 7)}\\=&\frac{x}{(x^{2} + x + 7)} + \frac{1}{2(x^{2} + x + 7)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{x}{(x^{2} + x + 7)} + \frac{1}{2(x^{2} + x + 7)}\right)}{dx}\\=&(\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}})x + \frac{1}{(x^{2} + x + 7)} + \frac{(\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}})}{2}\\=&\frac{-2x^{2}}{(x^{2} + x + 7)^{2}} - \frac{2x}{(x^{2} + x + 7)^{2}} + \frac{1}{(x^{2} + x + 7)} - \frac{1}{2(x^{2} + x + 7)^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2x^{2}}{(x^{2} + x + 7)^{2}} - \frac{2x}{(x^{2} + x + 7)^{2}} + \frac{1}{(x^{2} + x + 7)} - \frac{1}{2(x^{2} + x + 7)^{2}}\right)}{dx}\\=&-2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x^{2} - \frac{2*2x}{(x^{2} + x + 7)^{2}} - 2(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x - \frac{2}{(x^{2} + x + 7)^{2}} + (\frac{-(2x + 1 + 0)}{(x^{2} + x + 7)^{2}}) - \frac{(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})}{2}\\=&\frac{8x^{3}}{(x^{2} + x + 7)^{3}} + \frac{12x^{2}}{(x^{2} + x + 7)^{3}} - \frac{6x}{(x^{2} + x + 7)^{2}} + \frac{6x}{(x^{2} + x + 7)^{3}} - \frac{3}{(x^{2} + x + 7)^{2}} + \frac{1}{(x^{2} + x + 7)^{3}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{3}}{(x^{2} + x + 7)^{3}} + \frac{12x^{2}}{(x^{2} + x + 7)^{3}} - \frac{6x}{(x^{2} + x + 7)^{2}} + \frac{6x}{(x^{2} + x + 7)^{3}} - \frac{3}{(x^{2} + x + 7)^{2}} + \frac{1}{(x^{2} + x + 7)^{3}}\right)}{dx}\\=&8(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{3} + \frac{8*3x^{2}}{(x^{2} + x + 7)^{3}} + 12(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x^{2} + \frac{12*2x}{(x^{2} + x + 7)^{3}} - 6(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}})x - \frac{6}{(x^{2} + x + 7)^{2}} + 6(\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})x + \frac{6}{(x^{2} + x + 7)^{3}} - 3(\frac{-2(2x + 1 + 0)}{(x^{2} + x + 7)^{3}}) + (\frac{-3(2x + 1 + 0)}{(x^{2} + x + 7)^{4}})\\=&\frac{-48x^{4}}{(x^{2} + x + 7)^{4}} - \frac{96x^{3}}{(x^{2} + x + 7)^{4}} + \frac{48x^{2}}{(x^{2} + x + 7)^{3}} - \frac{72x^{2}}{(x^{2} + x + 7)^{4}} + \frac{48x}{(x^{2} + x + 7)^{3}} - \frac{24x}{(x^{2} + x + 7)^{4}} + \frac{12}{(x^{2} + x + 7)^{3}} - \frac{6}{(x^{2} + x + 7)^{2}} - \frac{3}{(x^{2} + x + 7)^{4}}\\ \end{split}\end{equation} \]

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