There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {log_{e^{x}}^{5}}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = {\left(log_{e^{x}}^{5}\right)}^{2}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( {\left(log_{e^{x}}^{5}\right)}^{2}\right)}{dx}\\=&(\frac{2log_{e^{x}}^{5}(\frac{(0)}{(5)} - \frac{(e^{x})log_{e^{x}}^{5}}{(e^{x})})}{(ln(e^{x}))})\\=&\frac{-2{\left(log_{e^{x}}^{5}\right)}^{2}}{ln(e^{x})}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2{\left(log_{e^{x}}^{5}\right)}^{2}}{ln(e^{x})}\right)}{dx}\\=&\frac{-2(\frac{2log_{e^{x}}^{5}(\frac{(0)}{(5)} - \frac{(e^{x})log_{e^{x}}^{5}}{(e^{x})})}{(ln(e^{x}))})}{ln(e^{x})} - \frac{2{\left(log_{e^{x}}^{5}\right)}^{2}*-e^{x}}{ln^{2}(e^{x})(e^{x})}\\=&\frac{6{\left(log_{e^{x}}^{5}\right)}^{2}}{ln^{2}(e^{x})}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{6{\left(log_{e^{x}}^{5}\right)}^{2}}{ln^{2}(e^{x})}\right)}{dx}\\=&\frac{6(\frac{2log_{e^{x}}^{5}(\frac{(0)}{(5)} - \frac{(e^{x})log_{e^{x}}^{5}}{(e^{x})})}{(ln(e^{x}))})}{ln^{2}(e^{x})} + \frac{6{\left(log_{e^{x}}^{5}\right)}^{2}*-2e^{x}}{ln^{3}(e^{x})(e^{x})}\\=&\frac{-24{\left(log_{e^{x}}^{5}\right)}^{2}}{ln^{3}(e^{x})}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-24{\left(log_{e^{x}}^{5}\right)}^{2}}{ln^{3}(e^{x})}\right)}{dx}\\=&\frac{-24(\frac{2log_{e^{x}}^{5}(\frac{(0)}{(5)} - \frac{(e^{x})log_{e^{x}}^{5}}{(e^{x})})}{(ln(e^{x}))})}{ln^{3}(e^{x})} - \frac{24{\left(log_{e^{x}}^{5}\right)}^{2}*-3e^{x}}{ln^{4}(e^{x})(e^{x})}\\=&\frac{120{\left(log_{e^{x}}^{5}\right)}^{2}}{ln^{4}(e^{x})}\\ \end{split}\end{equation} \]

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