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

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