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

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