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

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