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

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