There are 1 questions in this calculation: for each question, the 1 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ first\ derivative\ of\ function\ \frac{1}{5}{x}^{5} - 4sin(\frac{1}{2}x)(cos(\frac{1}{2})x)log_{5}^{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = - 4xlog_{5}^{x}sin(\frac{1}{2}x)cos(\frac{1}{2}) + \frac{1}{5}x^{5}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( - 4xlog_{5}^{x}sin(\frac{1}{2}x)cos(\frac{1}{2}) + \frac{1}{5}x^{5}\right)}{dx}\\=& - 4log_{5}^{x}sin(\frac{1}{2}x)cos(\frac{1}{2}) - 4x(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{5}^{x}}{(5)})}{(ln(5))})sin(\frac{1}{2}x)cos(\frac{1}{2}) - 4xlog_{5}^{x}cos(\frac{1}{2}x)*\frac{1}{2}cos(\frac{1}{2}) - 4xlog_{5}^{x}sin(\frac{1}{2}x)*-sin(\frac{1}{2})*0 + \frac{1}{5}*5x^{4}\\=& - 4log_{5}^{x}sin(\frac{1}{2}x)cos(\frac{1}{2}) - \frac{4sin(\frac{1}{2}x)cos(\frac{1}{2})}{ln(5)} - 2xlog_{5}^{x}cos(\frac{1}{2}x)cos(\frac{1}{2}) + x^{4}\\ \end{split}\end{equation} \]

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