There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg(log_{a}^{x})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg(log_{a}^{x})\right)}{dx}\\=&\frac{(\frac{(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{(ln(a))})}{ln{10}(log_{a}^{x})}\\=&\frac{1}{xlog(a, x)ln(a)ln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{1}{xlog(a, x)ln(a)ln{10}}\right)}{dx}\\=&\frac{-1}{x^{2}log(a, x)ln(a)ln{10}} + \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{xln(a)ln{10}} + \frac{-0}{xlog(a, x)ln^{2}(a)(a)ln{10}} + \frac{-0}{xlog(a, x)ln(a)ln^{2}{10}}\\=&\frac{-1}{x^{2}log(a, x)ln(a)ln{10}} - \frac{1}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{x^{2}log(a, x)ln(a)ln{10}} - \frac{1}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln{10}}\right)}{dx}\\=&\frac{--2}{x^{3}log(a, x)ln(a)ln{10}} - \frac{(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{2}ln(a)ln{10}} - \frac{-0}{x^{2}log(a, x)ln^{2}(a)(a)ln{10}} - \frac{-0}{x^{2}log(a, x)ln(a)ln^{2}{10}} - \frac{-2}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln{10}} - \frac{(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{2}ln^{2}(a)ln{10}} - \frac{-2*0}{x^{2}{\left(log(a, x)^{2}ln^{3}(a)(a)ln{10}} - \frac{-0}{x^{2}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}{10}}\\=&\frac{2}{x^{3}log(a, x)ln(a)ln{10}} + \frac{3}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln{10}} + \frac{2}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2}{x^{3}log(a, x)ln(a)ln{10}} + \frac{3}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln{10}} + \frac{2}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln{10}}\right)}{dx}\\=&\frac{2*-3}{x^{4}log(a, x)ln(a)ln{10}} + \frac{2(\frac{-(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{2}(ln(a))})}{x^{3}ln(a)ln{10}} + \frac{2*-0}{x^{3}log(a, x)ln^{2}(a)(a)ln{10}} + \frac{2*-0}{x^{3}log(a, x)ln(a)ln^{2}{10}} + \frac{3*-3}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln{10}} + \frac{3(\frac{-2(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{3}(ln(a))})}{x^{3}ln^{2}(a)ln{10}} + \frac{3*-2*0}{x^{3}{\left(log(a, x)^{2}ln^{3}(a)(a)ln{10}} + \frac{3*-0}{x^{3}{\left(log(a, x)^{2}ln^{2}(a)ln^{2}{10}} + \frac{2*-3}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln{10}} + \frac{2(\frac{-3(\frac{(1)}{(x)} - \frac{(0)log_{a}^{x}}{(a)})}{{\left(log(a, x)^{4}(ln(a))})}{x^{3}ln^{3}(a)ln{10}} + \frac{2*-3*0}{x^{3}{\left(log(a, x)^{3}ln^{4}(a)(a)ln{10}} + \frac{2*-0}{x^{3}{\left(log(a, x)^{3}ln^{3}(a)ln^{2}{10}}\\=&\frac{-6}{x^{4}log(a, x)ln(a)ln{10}} - \frac{11}{x^{4}{\left(log(a, x)^{2}ln^{2}(a)ln{10}} - \frac{12}{x^{4}{\left(log(a, x)^{3}ln^{3}(a)ln{10}} - \frac{6}{x^{4}{\left(log(a, x)^{4}ln^{4}(a)ln{10}}\\ \end{split}\end{equation} \]

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