There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ lg({x}^{lg(x)})\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( lg({x}^{lg(x)})\right)}{dx}\\=&\frac{({x}^{lg(x)}((\frac{1}{ln{10}(x)})ln(x) + \frac{(lg(x))(1)}{(x)}))}{ln{10}({x}^{lg(x)})}\\=&\frac{ln(x)}{xln^{2}{10}} + \frac{lg(x)}{xln{10}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{ln(x)}{xln^{2}{10}} + \frac{lg(x)}{xln{10}}\right)}{dx}\\=&\frac{-ln(x)}{x^{2}ln^{2}{10}} + \frac{-2*0ln(x)}{xln^{3}{10}} + \frac{1}{xln^{2}{10}(x)} + \frac{-lg(x)}{x^{2}ln{10}} + \frac{-0lg(x)}{xln^{2}{10}} + \frac{1}{xln{10}ln{10}(x)}\\=&\frac{-ln(x)}{x^{2}ln^{2}{10}} - \frac{lg(x)}{x^{2}ln{10}} + \frac{2}{x^{2}ln^{2}{10}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-ln(x)}{x^{2}ln^{2}{10}} - \frac{lg(x)}{x^{2}ln{10}} + \frac{2}{x^{2}ln^{2}{10}}\right)}{dx}\\=&\frac{--2ln(x)}{x^{3}ln^{2}{10}} - \frac{-2*0ln(x)}{x^{2}ln^{3}{10}} - \frac{1}{x^{2}ln^{2}{10}(x)} - \frac{-2lg(x)}{x^{3}ln{10}} - \frac{-0lg(x)}{x^{2}ln^{2}{10}} - \frac{1}{x^{2}ln{10}ln{10}(x)} + \frac{2*-2}{x^{3}ln^{2}{10}} + \frac{2*-2*0}{x^{2}ln^{3}{10}}\\=&\frac{2ln(x)}{x^{3}ln^{2}{10}} + \frac{2lg(x)}{x^{3}ln{10}} - \frac{6}{x^{3}ln^{2}{10}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{2ln(x)}{x^{3}ln^{2}{10}} + \frac{2lg(x)}{x^{3}ln{10}} - \frac{6}{x^{3}ln^{2}{10}}\right)}{dx}\\=&\frac{2*-3ln(x)}{x^{4}ln^{2}{10}} + \frac{2*-2*0ln(x)}{x^{3}ln^{3}{10}} + \frac{2}{x^{3}ln^{2}{10}(x)} + \frac{2*-3lg(x)}{x^{4}ln{10}} + \frac{2*-0lg(x)}{x^{3}ln^{2}{10}} + \frac{2}{x^{3}ln{10}ln{10}(x)} - \frac{6*-3}{x^{4}ln^{2}{10}} - \frac{6*-2*0}{x^{3}ln^{3}{10}}\\=&\frac{-6ln(x)}{x^{4}ln^{2}{10}} - \frac{6lg(x)}{x^{4}ln{10}} + \frac{22}{x^{4}ln^{2}{10}}\\ \end{split}\end{equation} \]

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