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

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