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

\[ \begin{equation}\begin{split}[2/2]Find\ the\ 4th\ derivative\ of\ function\ \frac{lg(x)}{x} + \frac{ln(x)}{(ln(10)x)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{lg(x)}{x} + \frac{ln(x)}{xln(10)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{lg(x)}{x} + \frac{ln(x)}{xln(10)}\right)}{dx}\\=&\frac{-lg(x)}{x^{2}} + \frac{1}{xln{10}(x)} + \frac{-ln(x)}{x^{2}ln(10)} + \frac{-0ln(x)}{xln^{2}(10)(10)} + \frac{1}{xln(10)(x)}\\=&\frac{-lg(x)}{x^{2}} - \frac{ln(x)}{x^{2}ln(10)} + \frac{1}{x^{2}ln{10}} + \frac{1}{x^{2}ln(10)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-lg(x)}{x^{2}} - \frac{ln(x)}{x^{2}ln(10)} + \frac{1}{x^{2}ln{10}} + \frac{1}{x^{2}ln(10)}\right)}{dx}\\=&\frac{--2lg(x)}{x^{3}} - \frac{1}{x^{2}ln{10}(x)} - \frac{-2ln(x)}{x^{3}ln(10)} - \frac{-0ln(x)}{x^{2}ln^{2}(10)(10)} - \frac{1}{x^{2}ln(10)(x)} + \frac{-2}{x^{3}ln{10}} + \frac{-0}{x^{2}ln^{2}{10}} + \frac{-2}{x^{3}ln(10)} + \frac{-0}{x^{2}ln^{2}(10)(10)}\\=&\frac{2lg(x)}{x^{3}} + \frac{2ln(x)}{x^{3}ln(10)} - \frac{3}{x^{3}ln{10}} - \frac{3}{x^{3}ln(10)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2lg(x)}{x^{3}} + \frac{2ln(x)}{x^{3}ln(10)} - \frac{3}{x^{3}ln{10}} - \frac{3}{x^{3}ln(10)}\right)}{dx}\\=&\frac{2*-3lg(x)}{x^{4}} + \frac{2}{x^{3}ln{10}(x)} + \frac{2*-3ln(x)}{x^{4}ln(10)} + \frac{2*-0ln(x)}{x^{3}ln^{2}(10)(10)} + \frac{2}{x^{3}ln(10)(x)} - \frac{3*-3}{x^{4}ln{10}} - \frac{3*-0}{x^{3}ln^{2}{10}} - \frac{3*-3}{x^{4}ln(10)} - \frac{3*-0}{x^{3}ln^{2}(10)(10)}\\=&\frac{-6lg(x)}{x^{4}} - \frac{6ln(x)}{x^{4}ln(10)} + \frac{11}{x^{4}ln{10}} + \frac{11}{x^{4}ln(10)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6lg(x)}{x^{4}} - \frac{6ln(x)}{x^{4}ln(10)} + \frac{11}{x^{4}ln{10}} + \frac{11}{x^{4}ln(10)}\right)}{dx}\\=&\frac{-6*-4lg(x)}{x^{5}} - \frac{6}{x^{4}ln{10}(x)} - \frac{6*-4ln(x)}{x^{5}ln(10)} - \frac{6*-0ln(x)}{x^{4}ln^{2}(10)(10)} - \frac{6}{x^{4}ln(10)(x)} + \frac{11*-4}{x^{5}ln{10}} + \frac{11*-0}{x^{4}ln^{2}{10}} + \frac{11*-4}{x^{5}ln(10)} + \frac{11*-0}{x^{4}ln^{2}(10)(10)}\\=&\frac{24lg(x)}{x^{5}} + \frac{24ln(x)}{x^{5}ln(10)} - \frac{50}{x^{5}ln{10}} - \frac{50}{x^{5}ln(10)}\\ \end{split}\end{equation} \]

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