There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
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\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ e^{e^{2lg(x)}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( e^{e^{2lg(x)}}\right)}{dx}\\=&\frac{e^{e^{2lg(x)}}e^{2lg(x)}*2}{ln{10}(x)}\\=&\frac{2e^{e^{2lg(x)}}e^{2lg(x)}}{xln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{2e^{e^{2lg(x)}}e^{2lg(x)}}{xln{10}}\right)}{dx}\\=&\frac{2*-e^{e^{2lg(x)}}e^{2lg(x)}}{x^{2}ln{10}} + \frac{2e^{e^{2lg(x)}}e^{2lg(x)}*2e^{2lg(x)}}{xln{10}(x)ln{10}} + \frac{2e^{e^{2lg(x)}}e^{2lg(x)}*2}{xln{10}(x)ln{10}} + \frac{2e^{e^{2lg(x)}}e^{2lg(x)}*-0}{xln^{2}{10}}\\=&\frac{-2e^{e^{2lg(x)}}e^{2lg(x)}}{x^{2}ln{10}} + \frac{4e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{2}ln^{2}{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}}{x^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2e^{e^{2lg(x)}}e^{2lg(x)}}{x^{2}ln{10}} + \frac{4e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{2}ln^{2}{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}}{x^{2}ln^{2}{10}}\right)}{dx}\\=&\frac{-2*-2e^{e^{2lg(x)}}e^{2lg(x)}}{x^{3}ln{10}} - \frac{2e^{e^{2lg(x)}}e^{2lg(x)}*2e^{2lg(x)}}{x^{2}ln{10}(x)ln{10}} - \frac{2e^{e^{2lg(x)}}e^{2lg(x)}*2}{x^{2}ln{10}(x)ln{10}} - \frac{2e^{e^{2lg(x)}}e^{2lg(x)}*-0}{x^{2}ln^{2}{10}} + \frac{4*-2e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{3}ln^{2}{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}*2e^{{2lg(x)}*{2}}}{x^{2}ln{10}(x)ln^{2}{10}} + \frac{4e^{e^{2lg(x)}}*2e^{2lg(x)}e^{2lg(x)}*2}{x^{2}ln{10}(x)ln^{2}{10}} + \frac{4e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}*-2*0}{x^{2}ln^{3}{10}} + \frac{4*-2e^{e^{2lg(x)}}e^{2lg(x)}}{x^{3}ln^{2}{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}*2e^{2lg(x)}}{x^{2}ln{10}(x)ln^{2}{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}*2}{x^{2}ln{10}(x)ln^{2}{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}*-2*0}{x^{2}ln^{3}{10}}\\=&\frac{4e^{e^{2lg(x)}}e^{2lg(x)}}{x^{3}ln{10}} - \frac{12e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{3}ln^{2}{10}} - \frac{12e^{e^{2lg(x)}}e^{2lg(x)}}{x^{3}ln^{2}{10}} + \frac{8e^{e^{2lg(x)}}e^{{2lg(x)}*{3}}}{x^{3}ln^{3}{10}} + \frac{16e^{{2lg(x)}*{2}}e^{e^{2lg(x)}}}{x^{3}ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{3}ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{2lg(x)}}{x^{3}ln^{3}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{4e^{e^{2lg(x)}}e^{2lg(x)}}{x^{3}ln{10}} - \frac{12e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{3}ln^{2}{10}} - \frac{12e^{e^{2lg(x)}}e^{2lg(x)}}{x^{3}ln^{2}{10}} + \frac{8e^{e^{2lg(x)}}e^{{2lg(x)}*{3}}}{x^{3}ln^{3}{10}} + \frac{16e^{{2lg(x)}*{2}}e^{e^{2lg(x)}}}{x^{3}ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{3}ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{2lg(x)}}{x^{3}ln^{3}{10}}\right)}{dx}\\=&\frac{4*-3e^{e^{2lg(x)}}e^{2lg(x)}}{x^{4}ln{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}*2e^{2lg(x)}}{x^{3}ln{10}(x)ln{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}*2}{x^{3}ln{10}(x)ln{10}} + \frac{4e^{e^{2lg(x)}}e^{2lg(x)}*-0}{x^{3}ln^{2}{10}} - \frac{12*-3e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{4}ln^{2}{10}} - \frac{12e^{e^{2lg(x)}}e^{2lg(x)}*2e^{{2lg(x)}*{2}}}{x^{3}ln{10}(x)ln^{2}{10}} - \frac{12e^{e^{2lg(x)}}*2e^{2lg(x)}e^{2lg(x)}*2}{x^{3}ln{10}(x)ln^{2}{10}} - \frac{12e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}*-2*0}{x^{3}ln^{3}{10}} - \frac{12*-3e^{e^{2lg(x)}}e^{2lg(x)}}{x^{4}ln^{2}{10}} - \frac{12e^{e^{2lg(x)}}e^{2lg(x)}*2e^{2lg(x)}}{x^{3}ln{10}(x)ln^{2}{10}} - \frac{12e^{e^{2lg(x)}}e^{2lg(x)}*2}{x^{3}ln{10}(x)ln^{2}{10}} - \frac{12e^{e^{2lg(x)}}e^{2lg(x)}*-2*0}{x^{3}ln^{3}{10}} + \frac{8*-3e^{e^{2lg(x)}}e^{{2lg(x)}*{3}}}{x^{4}ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{2lg(x)}*2e^{{2lg(x)}*{3}}}{x^{3}ln{10}(x)ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}*3e^{{2lg(x)}*{2}}e^{2lg(x)}*2}{x^{3}ln{10}(x)ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{{2lg(x)}*{3}}*-3*0}{x^{3}ln^{4}{10}} + \frac{16*-3e^{{2lg(x)}*{2}}e^{e^{2lg(x)}}}{x^{4}ln^{3}{10}} + \frac{16*2e^{2lg(x)}e^{2lg(x)}*2e^{e^{2lg(x)}}}{x^{3}ln{10}(x)ln^{3}{10}} + \frac{16e^{{2lg(x)}*{2}}e^{e^{2lg(x)}}e^{2lg(x)}*2}{x^{3}ln{10}(x)ln^{3}{10}} + \frac{16e^{{2lg(x)}*{2}}e^{e^{2lg(x)}}*-3*0}{x^{3}ln^{4}{10}} + \frac{8*-3e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{4}ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{2lg(x)}*2e^{{2lg(x)}*{2}}}{x^{3}ln{10}(x)ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}*2e^{2lg(x)}e^{2lg(x)}*2}{x^{3}ln{10}(x)ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}*-3*0}{x^{3}ln^{4}{10}} + \frac{8*-3e^{e^{2lg(x)}}e^{2lg(x)}}{x^{4}ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{2lg(x)}*2e^{2lg(x)}}{x^{3}ln{10}(x)ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{2lg(x)}*2}{x^{3}ln{10}(x)ln^{3}{10}} + \frac{8e^{e^{2lg(x)}}e^{2lg(x)}*-3*0}{x^{3}ln^{4}{10}}\\=&\frac{-12e^{e^{2lg(x)}}e^{2lg(x)}}{x^{4}ln{10}} + \frac{44e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{4}ln^{2}{10}} + \frac{44e^{e^{2lg(x)}}e^{2lg(x)}}{x^{4}ln^{2}{10}} - \frac{48e^{e^{2lg(x)}}e^{{2lg(x)}*{3}}}{x^{4}ln^{3}{10}} - \frac{96e^{{2lg(x)}*{2}}e^{e^{2lg(x)}}}{x^{4}ln^{3}{10}} - \frac{48e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{4}ln^{3}{10}} - \frac{48e^{e^{2lg(x)}}e^{2lg(x)}}{x^{4}ln^{3}{10}} + \frac{16e^{e^{2lg(x)}}e^{{2lg(x)}*{4}}}{x^{4}ln^{4}{10}} + \frac{80e^{{2lg(x)}*{3}}e^{e^{2lg(x)}}}{x^{4}ln^{4}{10}} + \frac{96e^{{2lg(x)}*{2}}e^{e^{2lg(x)}}}{x^{4}ln^{4}{10}} + \frac{16e^{e^{2lg(x)}}e^{{2lg(x)}*{3}}}{x^{4}ln^{4}{10}} + \frac{16e^{e^{2lg(x)}}e^{{2lg(x)}*{2}}}{x^{4}ln^{4}{10}} + \frac{16e^{e^{2lg(x)}}e^{2lg(x)}}{x^{4}ln^{4}{10}}\\ \end{split}\end{equation} \]

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