There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{ln(\frac{sin(x)}{x})}{x}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{ln(\frac{sin(x)}{x})}{x}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{ln(\frac{sin(x)}{x})}{x}\right)}{dx}\\=&\frac{-ln(\frac{sin(x)}{x})}{x^{2}} + \frac{(\frac{-sin(x)}{x^{2}} + \frac{cos(x)}{x})}{x(\frac{sin(x)}{x})}\\=&\frac{-ln(\frac{sin(x)}{x})}{x^{2}} + \frac{cos(x)}{xsin(x)} - \frac{1}{x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-ln(\frac{sin(x)}{x})}{x^{2}} + \frac{cos(x)}{xsin(x)} - \frac{1}{x^{2}}\right)}{dx}\\=&\frac{--2ln(\frac{sin(x)}{x})}{x^{3}} - \frac{(\frac{-sin(x)}{x^{2}} + \frac{cos(x)}{x})}{x^{2}(\frac{sin(x)}{x})} + \frac{-cos(x)}{x^{2}sin(x)} + \frac{-cos(x)cos(x)}{xsin^{2}(x)} + \frac{-sin(x)}{xsin(x)} - \frac{-2}{x^{3}}\\=&\frac{2ln(\frac{sin(x)}{x})}{x^{3}} - \frac{2cos(x)}{x^{2}sin(x)} - \frac{cos^{2}(x)}{xsin^{2}(x)} + \frac{3}{x^{3}} - \frac{1}{x}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{2ln(\frac{sin(x)}{x})}{x^{3}} - \frac{2cos(x)}{x^{2}sin(x)} - \frac{cos^{2}(x)}{xsin^{2}(x)} + \frac{3}{x^{3}} - \frac{1}{x}\right)}{dx}\\=&\frac{2*-3ln(\frac{sin(x)}{x})}{x^{4}} + \frac{2(\frac{-sin(x)}{x^{2}} + \frac{cos(x)}{x})}{x^{3}(\frac{sin(x)}{x})} - \frac{2*-2cos(x)}{x^{3}sin(x)} - \frac{2*-cos(x)cos(x)}{x^{2}sin^{2}(x)} - \frac{2*-sin(x)}{x^{2}sin(x)} - \frac{-cos^{2}(x)}{x^{2}sin^{2}(x)} - \frac{-2cos(x)cos^{2}(x)}{xsin^{3}(x)} - \frac{-2cos(x)sin(x)}{xsin^{2}(x)} + \frac{3*-3}{x^{4}} - \frac{-1}{x^{2}}\\=&\frac{-6ln(\frac{sin(x)}{x})}{x^{4}} + \frac{6cos(x)}{x^{3}sin(x)} + \frac{3cos^{2}(x)}{x^{2}sin^{2}(x)} + \frac{2cos(x)}{xsin(x)} + \frac{2cos^{3}(x)}{xsin^{3}(x)} - \frac{11}{x^{4}} + \frac{3}{x^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-6ln(\frac{sin(x)}{x})}{x^{4}} + \frac{6cos(x)}{x^{3}sin(x)} + \frac{3cos^{2}(x)}{x^{2}sin^{2}(x)} + \frac{2cos(x)}{xsin(x)} + \frac{2cos^{3}(x)}{xsin^{3}(x)} - \frac{11}{x^{4}} + \frac{3}{x^{2}}\right)}{dx}\\=&\frac{-6*-4ln(\frac{sin(x)}{x})}{x^{5}} - \frac{6(\frac{-sin(x)}{x^{2}} + \frac{cos(x)}{x})}{x^{4}(\frac{sin(x)}{x})} + \frac{6*-3cos(x)}{x^{4}sin(x)} + \frac{6*-cos(x)cos(x)}{x^{3}sin^{2}(x)} + \frac{6*-sin(x)}{x^{3}sin(x)} + \frac{3*-2cos^{2}(x)}{x^{3}sin^{2}(x)} + \frac{3*-2cos(x)cos^{2}(x)}{x^{2}sin^{3}(x)} + \frac{3*-2cos(x)sin(x)}{x^{2}sin^{2}(x)} + \frac{2*-cos(x)}{x^{2}sin(x)} + \frac{2*-cos(x)cos(x)}{xsin^{2}(x)} + \frac{2*-sin(x)}{xsin(x)} + \frac{2*-cos^{3}(x)}{x^{2}sin^{3}(x)} + \frac{2*-3cos(x)cos^{3}(x)}{xsin^{4}(x)} + \frac{2*-3cos^{2}(x)sin(x)}{xsin^{3}(x)} - \frac{11*-4}{x^{5}} + \frac{3*-2}{x^{3}}\\=&\frac{24ln(\frac{sin(x)}{x})}{x^{5}} - \frac{24cos(x)}{x^{4}sin(x)} - \frac{12cos^{2}(x)}{x^{3}sin^{2}(x)} - \frac{8cos(x)}{x^{2}sin(x)} - \frac{8cos^{3}(x)}{x^{2}sin^{3}(x)} - \frac{8cos^{2}(x)}{xsin^{2}(x)} - \frac{6cos^{4}(x)}{xsin^{4}(x)} - \frac{2}{x} + \frac{50}{x^{5}} - \frac{12}{x^{3}}\\ \end{split}\end{equation} \]

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