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

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