There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ {({e}^{\frac{1}{x}}{(x{(sin(x))}^{\frac{1}{2}})}^{\frac{1}{2}})}^{\frac{1}{2}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = x^{\frac{1}{4}}{e}^{(\frac{\frac{1}{2}}{x})}sin^{\frac{1}{8}}(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( x^{\frac{1}{4}}{e}^{(\frac{\frac{1}{2}}{x})}sin^{\frac{1}{8}}(x)\right)}{dx}\\=&\frac{\frac{1}{4}{e}^{(\frac{\frac{1}{2}}{x})}sin^{\frac{1}{8}}(x)}{x^{\frac{3}{4}}} + x^{\frac{1}{4}}({e}^{(\frac{\frac{1}{2}}{x})}((\frac{\frac{1}{2}*-1}{x^{2}})ln(e) + \frac{(\frac{\frac{1}{2}}{x})(0)}{(e)}))sin^{\frac{1}{8}}(x) + \frac{x^{\frac{1}{4}}{e}^{(\frac{\frac{1}{2}}{x})}*\frac{1}{8}cos(x)}{sin^{\frac{7}{8}}(x)}\\=&\frac{x^{\frac{1}{4}}{e}^{(\frac{\frac{1}{2}}{x})}cos(x)}{8sin^{\frac{7}{8}}(x)} - \frac{{e}^{(\frac{\frac{1}{2}}{x})}sin^{\frac{1}{8}}(x)}{2x^{\frac{7}{4}}} + \frac{{e}^{(\frac{\frac{1}{2}}{x})}sin^{\frac{1}{8}}(x)}{4x^{\frac{3}{4}}}\\ \end{split}\end{equation} \]

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