There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{({x}^{\frac{1}{2}} + 4)}{(-{x}^{3} + 23x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{\frac{1}{2}}}{(-x^{3} + 23x)} + \frac{4}{(-x^{3} + 23x)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{\frac{1}{2}}}{(-x^{3} + 23x)} + \frac{4}{(-x^{3} + 23x)}\right)}{dx}\\=&(\frac{-(-3x^{2} + 23)}{(-x^{3} + 23x)^{2}})x^{\frac{1}{2}} + \frac{\frac{1}{2}}{(-x^{3} + 23x)x^{\frac{1}{2}}} + 4(\frac{-(-3x^{2} + 23)}{(-x^{3} + 23x)^{2}})\\=&\frac{3x^{\frac{5}{2}}}{(-x^{3} + 23x)^{2}} - \frac{23x^{\frac{1}{2}}}{(-x^{3} + 23x)^{2}} + \frac{1}{2(-x^{3} + 23x)x^{\frac{1}{2}}} + \frac{12x^{2}}{(-x^{3} + 23x)^{2}} - \frac{92}{(-x^{3} + 23x)^{2}}\\ \end{split}\end{equation} \]

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