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

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