There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{{(\frac{{e}^{\frac{1}{x}}{(\frac{x{(sin(x))}^{1}}{2})}^{1}}{2})}^{1}}{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{8}x{e}^{\frac{1}{x}}sin(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{8}x{e}^{\frac{1}{x}}sin(x)\right)}{dx}\\=&\frac{1}{8}{e}^{\frac{1}{x}}sin(x) + \frac{1}{8}x({e}^{\frac{1}{x}}((\frac{-1}{x^{2}})ln(e) + \frac{(\frac{1}{x})(0)}{(e)}))sin(x) + \frac{1}{8}x{e}^{\frac{1}{x}}cos(x)\\=&\frac{{e}^{\frac{1}{x}}sin(x)}{8} - \frac{{e}^{\frac{1}{x}}sin(x)}{8x} + \frac{x{e}^{\frac{1}{x}}cos(x)}{8}\\ \end{split}\end{equation} \]

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