There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{({x}^{2} + 1)}{({x}^{4} + 1)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x^{2}}{(x^{4} + 1)} + \frac{1}{(x^{4} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x^{2}}{(x^{4} + 1)} + \frac{1}{(x^{4} + 1)}\right)}{dx}\\=&(\frac{-(4x^{3} + 0)}{(x^{4} + 1)^{2}})x^{2} + \frac{2x}{(x^{4} + 1)} + (\frac{-(4x^{3} + 0)}{(x^{4} + 1)^{2}})\\=&\frac{-4x^{5}}{(x^{4} + 1)^{2}} + \frac{2x}{(x^{4} + 1)} - \frac{4x^{3}}{(x^{4} + 1)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-4x^{5}}{(x^{4} + 1)^{2}} + \frac{2x}{(x^{4} + 1)} - \frac{4x^{3}}{(x^{4} + 1)^{2}}\right)}{dx}\\=&-4(\frac{-2(4x^{3} + 0)}{(x^{4} + 1)^{3}})x^{5} - \frac{4*5x^{4}}{(x^{4} + 1)^{2}} + 2(\frac{-(4x^{3} + 0)}{(x^{4} + 1)^{2}})x + \frac{2}{(x^{4} + 1)} - 4(\frac{-2(4x^{3} + 0)}{(x^{4} + 1)^{3}})x^{3} - \frac{4*3x^{2}}{(x^{4} + 1)^{2}}\\=&\frac{32x^{8}}{(x^{4} + 1)^{3}} - \frac{28x^{4}}{(x^{4} + 1)^{2}} + \frac{32x^{6}}{(x^{4} + 1)^{3}} - \frac{12x^{2}}{(x^{4} + 1)^{2}} + \frac{2}{(x^{4} + 1)}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{32x^{8}}{(x^{4} + 1)^{3}} - \frac{28x^{4}}{(x^{4} + 1)^{2}} + \frac{32x^{6}}{(x^{4} + 1)^{3}} - \frac{12x^{2}}{(x^{4} + 1)^{2}} + \frac{2}{(x^{4} + 1)}\right)}{dx}\\=&32(\frac{-3(4x^{3} + 0)}{(x^{4} + 1)^{4}})x^{8} + \frac{32*8x^{7}}{(x^{4} + 1)^{3}} - 28(\frac{-2(4x^{3} + 0)}{(x^{4} + 1)^{3}})x^{4} - \frac{28*4x^{3}}{(x^{4} + 1)^{2}} + 32(\frac{-3(4x^{3} + 0)}{(x^{4} + 1)^{4}})x^{6} + \frac{32*6x^{5}}{(x^{4} + 1)^{3}} - 12(\frac{-2(4x^{3} + 0)}{(x^{4} + 1)^{3}})x^{2} - \frac{12*2x}{(x^{4} + 1)^{2}} + 2(\frac{-(4x^{3} + 0)}{(x^{4} + 1)^{2}})\\=&\frac{-384x^{11}}{(x^{4} + 1)^{4}} + \frac{480x^{7}}{(x^{4} + 1)^{3}} - \frac{120x^{3}}{(x^{4} + 1)^{2}} - \frac{384x^{9}}{(x^{4} + 1)^{4}} + \frac{288x^{5}}{(x^{4} + 1)^{3}} - \frac{24x}{(x^{4} + 1)^{2}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-384x^{11}}{(x^{4} + 1)^{4}} + \frac{480x^{7}}{(x^{4} + 1)^{3}} - \frac{120x^{3}}{(x^{4} + 1)^{2}} - \frac{384x^{9}}{(x^{4} + 1)^{4}} + \frac{288x^{5}}{(x^{4} + 1)^{3}} - \frac{24x}{(x^{4} + 1)^{2}}\right)}{dx}\\=&-384(\frac{-4(4x^{3} + 0)}{(x^{4} + 1)^{5}})x^{11} - \frac{384*11x^{10}}{(x^{4} + 1)^{4}} + 480(\frac{-3(4x^{3} + 0)}{(x^{4} + 1)^{4}})x^{7} + \frac{480*7x^{6}}{(x^{4} + 1)^{3}} - 120(\frac{-2(4x^{3} + 0)}{(x^{4} + 1)^{3}})x^{3} - \frac{120*3x^{2}}{(x^{4} + 1)^{2}} - 384(\frac{-4(4x^{3} + 0)}{(x^{4} + 1)^{5}})x^{9} - \frac{384*9x^{8}}{(x^{4} + 1)^{4}} + 288(\frac{-3(4x^{3} + 0)}{(x^{4} + 1)^{4}})x^{5} + \frac{288*5x^{4}}{(x^{4} + 1)^{3}} - 24(\frac{-2(4x^{3} + 0)}{(x^{4} + 1)^{3}})x - \frac{24}{(x^{4} + 1)^{2}}\\=&\frac{6144x^{14}}{(x^{4} + 1)^{5}} - \frac{9984x^{10}}{(x^{4} + 1)^{4}} + \frac{4320x^{6}}{(x^{4} + 1)^{3}} - \frac{360x^{2}}{(x^{4} + 1)^{2}} + \frac{6144x^{12}}{(x^{4} + 1)^{5}} - \frac{6912x^{8}}{(x^{4} + 1)^{4}} + \frac{1632x^{4}}{(x^{4} + 1)^{3}} - \frac{24}{(x^{4} + 1)^{2}}\\ \end{split}\end{equation} \]

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