There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{{x}^{4}}{(1 + {x}^{3})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{4}}{(x^{3} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{4}}{(x^{3} + 1)}\right)}{dx}\\=&(\frac{-(3x^{2} + 0)}{(x^{3} + 1)^{2}})x^{4} + \frac{4x^{3}}{(x^{3} + 1)}\\=&\frac{-3x^{6}}{(x^{3} + 1)^{2}} + \frac{4x^{3}}{(x^{3} + 1)}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-3x^{6}}{(x^{3} + 1)^{2}} + \frac{4x^{3}}{(x^{3} + 1)}\right)}{dx}\\=&-3(\frac{-2(3x^{2} + 0)}{(x^{3} + 1)^{3}})x^{6} - \frac{3*6x^{5}}{(x^{3} + 1)^{2}} + 4(\frac{-(3x^{2} + 0)}{(x^{3} + 1)^{2}})x^{3} + \frac{4*3x^{2}}{(x^{3} + 1)}\\=&\frac{18x^{8}}{(x^{3} + 1)^{3}} - \frac{30x^{5}}{(x^{3} + 1)^{2}} + \frac{12x^{2}}{(x^{3} + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{18x^{8}}{(x^{3} + 1)^{3}} - \frac{30x^{5}}{(x^{3} + 1)^{2}} + \frac{12x^{2}}{(x^{3} + 1)}\right)}{dx}\\=&18(\frac{-3(3x^{2} + 0)}{(x^{3} + 1)^{4}})x^{8} + \frac{18*8x^{7}}{(x^{3} + 1)^{3}} - 30(\frac{-2(3x^{2} + 0)}{(x^{3} + 1)^{3}})x^{5} - \frac{30*5x^{4}}{(x^{3} + 1)^{2}} + 12(\frac{-(3x^{2} + 0)}{(x^{3} + 1)^{2}})x^{2} + \frac{12*2x}{(x^{3} + 1)}\\=&\frac{-162x^{10}}{(x^{3} + 1)^{4}} + \frac{324x^{7}}{(x^{3} + 1)^{3}} - \frac{186x^{4}}{(x^{3} + 1)^{2}} + \frac{24x}{(x^{3} + 1)}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-162x^{10}}{(x^{3} + 1)^{4}} + \frac{324x^{7}}{(x^{3} + 1)^{3}} - \frac{186x^{4}}{(x^{3} + 1)^{2}} + \frac{24x}{(x^{3} + 1)}\right)}{dx}\\=&-162(\frac{-4(3x^{2} + 0)}{(x^{3} + 1)^{5}})x^{10} - \frac{162*10x^{9}}{(x^{3} + 1)^{4}} + 324(\frac{-3(3x^{2} + 0)}{(x^{3} + 1)^{4}})x^{7} + \frac{324*7x^{6}}{(x^{3} + 1)^{3}} - 186(\frac{-2(3x^{2} + 0)}{(x^{3} + 1)^{3}})x^{4} - \frac{186*4x^{3}}{(x^{3} + 1)^{2}} + 24(\frac{-(3x^{2} + 0)}{(x^{3} + 1)^{2}})x + \frac{24}{(x^{3} + 1)}\\=&\frac{1944x^{12}}{(x^{3} + 1)^{5}} - \frac{4536x^{9}}{(x^{3} + 1)^{4}} + \frac{3384x^{6}}{(x^{3} + 1)^{3}} - \frac{816x^{3}}{(x^{3} + 1)^{2}} + \frac{24}{(x^{3} + 1)}\\ \end{split}\end{equation} \]

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