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

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