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

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