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

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