There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ ln(x){\frac{1}{x}}^{2} - (ln(\frac{e}{x})){\frac{1}{(\frac{e}{x})}}^{2}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{ln(x)}{x^{2}} - \frac{x^{2}ln(\frac{e}{x})}{e^{2}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ln(x)}{x^{2}} - \frac{x^{2}ln(\frac{e}{x})}{e^{2}}\right)}{dx}\\=&\frac{-2ln(x)}{x^{3}} + \frac{1}{x^{2}(x)} - \frac{2xln(\frac{e}{x})}{e^{2}} - \frac{x^{2}*-2*0ln(\frac{e}{x})}{e^{3}} - \frac{x^{2}(\frac{-e}{x^{2}} + \frac{0}{x})}{e^{2}(\frac{e}{x})}\\=&\frac{-2ln(x)}{x^{3}} - \frac{2xln(\frac{e}{x})}{e^{2}} + \frac{x}{e^{2}} + \frac{1}{x^{3}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2ln(x)}{x^{3}} - \frac{2xln(\frac{e}{x})}{e^{2}} + \frac{x}{e^{2}} + \frac{1}{x^{3}}\right)}{dx}\\=&\frac{-2*-3ln(x)}{x^{4}} - \frac{2}{x^{3}(x)} - \frac{2ln(\frac{e}{x})}{e^{2}} - \frac{2x*-2*0ln(\frac{e}{x})}{e^{3}} - \frac{2x(\frac{-e}{x^{2}} + \frac{0}{x})}{e^{2}(\frac{e}{x})} + \frac{1}{e^{2}} + \frac{x*-2*0}{e^{3}} + \frac{-3}{x^{4}}\\=&\frac{6ln(x)}{x^{4}} - \frac{5}{x^{4}} - \frac{2ln(\frac{e}{x})}{e^{2}} + \frac{3}{e^{2}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{6ln(x)}{x^{4}} - \frac{5}{x^{4}} - \frac{2ln(\frac{e}{x})}{e^{2}} + \frac{3}{e^{2}}\right)}{dx}\\=&\frac{6*-4ln(x)}{x^{5}} + \frac{6}{x^{4}(x)} - \frac{5*-4}{x^{5}} - \frac{2*-2*0ln(\frac{e}{x})}{e^{3}} - \frac{2(\frac{-e}{x^{2}} + \frac{0}{x})}{e^{2}(\frac{e}{x})} + \frac{3*-2*0}{e^{3}}\\=&\frac{-24ln(x)}{x^{5}} + \frac{2}{xe^{2}} + \frac{26}{x^{5}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-24ln(x)}{x^{5}} + \frac{2}{xe^{2}} + \frac{26}{x^{5}}\right)}{dx}\\=&\frac{-24*-5ln(x)}{x^{6}} - \frac{24}{x^{5}(x)} + \frac{2*-1}{x^{2}e^{2}} + \frac{2*-2*0}{xe^{3}} + \frac{26*-5}{x^{6}}\\=&\frac{120ln(x)}{x^{6}} - \frac{2}{x^{2}e^{2}} - \frac{154}{x^{6}}\\ \end{split}\end{equation} \]

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