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

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