There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{(\frac{-2{(1 + x)}^{1}}{2} + \frac{{(-8(\frac{{x}^{1}}{2})ln(x))}^{1}}{2})ln(x)}{4}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = - \frac{1}{4}xln(x) - \frac{1}{4}ln(x) - \frac{1}{2}xln^{2}(x)\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( - \frac{1}{4}xln(x) - \frac{1}{4}ln(x) - \frac{1}{2}xln^{2}(x)\right)}{dx}\\=& - \frac{1}{4}ln(x) - \frac{\frac{1}{4}x}{(x)} - \frac{\frac{1}{4}}{(x)} - \frac{1}{2}ln^{2}(x) - \frac{\frac{1}{2}x*2ln(x)}{(x)}\\=& - \frac{5ln(x)}{4} - \frac{1}{4x} - \frac{ln^{2}(x)}{2} - \frac{1}{4}\\ \end{split}\end{equation} \]

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