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

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