There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{(1 + x + {x}^{2})}{(1 - x + {x}^{2})}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{x}{(-x + x^{2} + 1)} + \frac{x^{2}}{(-x + x^{2} + 1)} + \frac{1}{(-x + x^{2} + 1)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{x}{(-x + x^{2} + 1)} + \frac{x^{2}}{(-x + x^{2} + 1)} + \frac{1}{(-x + x^{2} + 1)}\right)}{dx}\\=&(\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}})x + \frac{1}{(-x + x^{2} + 1)} + (\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}})x^{2} + \frac{2x}{(-x + x^{2} + 1)} + (\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}})\\=& - \frac{x^{2}}{(-x + x^{2} + 1)^{2}} - \frac{2x^{3}}{(-x + x^{2} + 1)^{2}} + \frac{2x}{(-x + x^{2} + 1)} - \frac{x}{(-x + x^{2} + 1)^{2}} + \frac{1}{(-x + x^{2} + 1)} + \frac{1}{(-x + x^{2} + 1)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( - \frac{x^{2}}{(-x + x^{2} + 1)^{2}} - \frac{2x^{3}}{(-x + x^{2} + 1)^{2}} + \frac{2x}{(-x + x^{2} + 1)} - \frac{x}{(-x + x^{2} + 1)^{2}} + \frac{1}{(-x + x^{2} + 1)} + \frac{1}{(-x + x^{2} + 1)^{2}}\right)}{dx}\\=& - (\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x^{2} - \frac{2x}{(-x + x^{2} + 1)^{2}} - 2(\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x^{3} - \frac{2*3x^{2}}{(-x + x^{2} + 1)^{2}} + 2(\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}})x + \frac{2}{(-x + x^{2} + 1)} - (\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x - \frac{1}{(-x + x^{2} + 1)^{2}} + (\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}}) + (\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})\\=&\frac{8x^{4}}{(-x + x^{2} + 1)^{3}} - \frac{2x}{(-x + x^{2} + 1)^{2}} + \frac{2x^{2}}{(-x + x^{2} + 1)^{3}} - \frac{10x^{2}}{(-x + x^{2} + 1)^{2}} - \frac{6x}{(-x + x^{2} + 1)^{3}} + \frac{2}{(-x + x^{2} + 1)} + \frac{2}{(-x + x^{2} + 1)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8x^{4}}{(-x + x^{2} + 1)^{3}} - \frac{2x}{(-x + x^{2} + 1)^{2}} + \frac{2x^{2}}{(-x + x^{2} + 1)^{3}} - \frac{10x^{2}}{(-x + x^{2} + 1)^{2}} - \frac{6x}{(-x + x^{2} + 1)^{3}} + \frac{2}{(-x + x^{2} + 1)} + \frac{2}{(-x + x^{2} + 1)^{3}}\right)}{dx}\\=&8(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x^{4} + \frac{8*4x^{3}}{(-x + x^{2} + 1)^{3}} - 2(\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x - \frac{2}{(-x + x^{2} + 1)^{2}} + 2(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x^{2} + \frac{2*2x}{(-x + x^{2} + 1)^{3}} - 10(\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x^{2} - \frac{10*2x}{(-x + x^{2} + 1)^{2}} - 6(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x - \frac{6}{(-x + x^{2} + 1)^{3}} + 2(\frac{-(-1 + 2x + 0)}{(-x + x^{2} + 1)^{2}}) + 2(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})\\=&\frac{-48x^{5}}{(-x + x^{2} + 1)^{4}} - \frac{12x^{3}}{(-x + x^{2} + 1)^{4}} + \frac{72x^{3}}{(-x + x^{2} + 1)^{3}} - \frac{12x^{2}}{(-x + x^{2} + 1)^{3}} + \frac{42x^{2}}{(-x + x^{2} + 1)^{4}} - \frac{24x}{(-x + x^{2} + 1)^{2}} + \frac{24x^{4}}{(-x + x^{2} + 1)^{4}} - \frac{30x}{(-x + x^{2} + 1)^{4}} - \frac{6}{(-x + x^{2} + 1)^{3}} + \frac{6}{(-x + x^{2} + 1)^{4}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-48x^{5}}{(-x + x^{2} + 1)^{4}} - \frac{12x^{3}}{(-x + x^{2} + 1)^{4}} + \frac{72x^{3}}{(-x + x^{2} + 1)^{3}} - \frac{12x^{2}}{(-x + x^{2} + 1)^{3}} + \frac{42x^{2}}{(-x + x^{2} + 1)^{4}} - \frac{24x}{(-x + x^{2} + 1)^{2}} + \frac{24x^{4}}{(-x + x^{2} + 1)^{4}} - \frac{30x}{(-x + x^{2} + 1)^{4}} - \frac{6}{(-x + x^{2} + 1)^{3}} + \frac{6}{(-x + x^{2} + 1)^{4}}\right)}{dx}\\=&-48(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x^{5} - \frac{48*5x^{4}}{(-x + x^{2} + 1)^{4}} - 12(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x^{3} - \frac{12*3x^{2}}{(-x + x^{2} + 1)^{4}} + 72(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x^{3} + \frac{72*3x^{2}}{(-x + x^{2} + 1)^{3}} - 12(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}})x^{2} - \frac{12*2x}{(-x + x^{2} + 1)^{3}} + 42(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x^{2} + \frac{42*2x}{(-x + x^{2} + 1)^{4}} - 24(\frac{-2(-1 + 2x + 0)}{(-x + x^{2} + 1)^{3}})x - \frac{24}{(-x + x^{2} + 1)^{2}} + 24(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x^{4} + \frac{24*4x^{3}}{(-x + x^{2} + 1)^{4}} - 30(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})x - \frac{30}{(-x + x^{2} + 1)^{4}} - 6(\frac{-3(-1 + 2x + 0)}{(-x + x^{2} + 1)^{4}}) + 6(\frac{-4(-1 + 2x + 0)}{(-x + x^{2} + 1)^{5}})\\=&\frac{384x^{6}}{(-x + x^{2} + 1)^{5}} + \frac{192x^{4}}{(-x + x^{2} + 1)^{5}} - \frac{672x^{4}}{(-x + x^{2} + 1)^{4}} - \frac{384x^{3}}{(-x + x^{2} + 1)^{5}} - \frac{384x^{5}}{(-x + x^{2} + 1)^{5}} - \frac{72x^{2}}{(-x + x^{2} + 1)^{4}} + \frac{384x^{3}}{(-x + x^{2} + 1)^{4}} + \frac{312x^{2}}{(-x + x^{2} + 1)^{3}} - \frac{72x}{(-x + x^{2} + 1)^{3}} + \frac{408x^{2}}{(-x + x^{2} + 1)^{5}} + \frac{120x}{(-x + x^{2} + 1)^{4}} - \frac{168x}{(-x + x^{2} + 1)^{5}} - \frac{48}{(-x + x^{2} + 1)^{4}} - \frac{24}{(-x + x^{2} + 1)^{2}} + \frac{24}{(-x + x^{2} + 1)^{5}}\\ \end{split}\end{equation} \]

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