There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ {(1 + \frac{3}{x})}^{\frac{1}{3}}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = (\frac{3}{x} + 1)^{\frac{1}{3}}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( (\frac{3}{x} + 1)^{\frac{1}{3}}\right)}{dx}\\=&(\frac{\frac{1}{3}(\frac{3*-1}{x^{2}} + 0)}{(\frac{3}{x} + 1)^{\frac{2}{3}}})\\=&\frac{-1}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-1}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{2}}\right)}{dx}\\=&\frac{-(\frac{\frac{-2}{3}(\frac{3*-1}{x^{2}} + 0)}{(\frac{3}{x} + 1)^{\frac{5}{3}}})}{x^{2}} - \frac{-2}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{3}}\\=&\frac{-2}{(\frac{3}{x} + 1)^{\frac{5}{3}}x^{4}} + \frac{2}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2}{(\frac{3}{x} + 1)^{\frac{5}{3}}x^{4}} + \frac{2}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{3}}\right)}{dx}\\=&\frac{-2(\frac{\frac{-5}{3}(\frac{3*-1}{x^{2}} + 0)}{(\frac{3}{x} + 1)^{\frac{8}{3}}})}{x^{4}} - \frac{2*-4}{(\frac{3}{x} + 1)^{\frac{5}{3}}x^{5}} + \frac{2(\frac{\frac{-2}{3}(\frac{3*-1}{x^{2}} + 0)}{(\frac{3}{x} + 1)^{\frac{5}{3}}})}{x^{3}} + \frac{2*-3}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{4}}\\=&\frac{-10}{(\frac{3}{x} + 1)^{\frac{8}{3}}x^{6}} + \frac{12}{(\frac{3}{x} + 1)^{\frac{5}{3}}x^{5}} - \frac{6}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{4}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-10}{(\frac{3}{x} + 1)^{\frac{8}{3}}x^{6}} + \frac{12}{(\frac{3}{x} + 1)^{\frac{5}{3}}x^{5}} - \frac{6}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{4}}\right)}{dx}\\=&\frac{-10(\frac{\frac{-8}{3}(\frac{3*-1}{x^{2}} + 0)}{(\frac{3}{x} + 1)^{\frac{11}{3}}})}{x^{6}} - \frac{10*-6}{(\frac{3}{x} + 1)^{\frac{8}{3}}x^{7}} + \frac{12(\frac{\frac{-5}{3}(\frac{3*-1}{x^{2}} + 0)}{(\frac{3}{x} + 1)^{\frac{8}{3}}})}{x^{5}} + \frac{12*-5}{(\frac{3}{x} + 1)^{\frac{5}{3}}x^{6}} - \frac{6(\frac{\frac{-2}{3}(\frac{3*-1}{x^{2}} + 0)}{(\frac{3}{x} + 1)^{\frac{5}{3}}})}{x^{4}} - \frac{6*-4}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{5}}\\=&\frac{-80}{(\frac{3}{x} + 1)^{\frac{11}{3}}x^{8}} + \frac{120}{(\frac{3}{x} + 1)^{\frac{8}{3}}x^{7}} - \frac{72}{(\frac{3}{x} + 1)^{\frac{5}{3}}x^{6}} + \frac{24}{(\frac{3}{x} + 1)^{\frac{2}{3}}x^{5}}\\ \end{split}\end{equation} \]

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