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

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