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

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