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

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