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

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