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

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