There are 1 questions in this calculation: for each question, the 4 derivative of x is calculated.
    Note that variables are case sensitive.
\[ \begin{equation}\begin{split}[1/1]Find\ the\ 4th\ derivative\ of\ function\ \frac{1}{(a{x}^{2} + bx + c)}\ with\ respect\ to\ x：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\Solution:&\\ &Primitive\ function\ = \frac{1}{(ax^{2} + bx + c)}\\&\color{blue}{The\ first\ derivative\ function：}\\&\frac{d\left( \frac{1}{(ax^{2} + bx + c)}\right)}{dx}\\=&(\frac{-(a*2x + b + 0)}{(ax^{2} + bx + c)^{2}})\\=&\frac{-2ax}{(ax^{2} + bx + c)^{2}} - \frac{b}{(ax^{2} + bx + c)^{2}}\\\\ &\color{blue}{The\ second\ derivative\ of\ function:} \\&\frac{d\left( \frac{-2ax}{(ax^{2} + bx + c)^{2}} - \frac{b}{(ax^{2} + bx + c)^{2}}\right)}{dx}\\=&-2(\frac{-2(a*2x + b + 0)}{(ax^{2} + bx + c)^{3}})ax - \frac{2a}{(ax^{2} + bx + c)^{2}} - (\frac{-2(a*2x + b + 0)}{(ax^{2} + bx + c)^{3}})b + 0\\=&\frac{8a^{2}x^{2}}{(ax^{2} + bx + c)^{3}} + \frac{8abx}{(ax^{2} + bx + c)^{3}} - \frac{2a}{(ax^{2} + bx + c)^{2}} + \frac{2b^{2}}{(ax^{2} + bx + c)^{3}}\\\\ &\color{blue}{The\ third\ derivative\ of\ function:} \\&\frac{d\left( \frac{8a^{2}x^{2}}{(ax^{2} + bx + c)^{3}} + \frac{8abx}{(ax^{2} + bx + c)^{3}} - \frac{2a}{(ax^{2} + bx + c)^{2}} + \frac{2b^{2}}{(ax^{2} + bx + c)^{3}}\right)}{dx}\\=&8(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})a^{2}x^{2} + \frac{8a^{2}*2x}{(ax^{2} + bx + c)^{3}} + 8(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})abx + \frac{8ab}{(ax^{2} + bx + c)^{3}} - 2(\frac{-2(a*2x + b + 0)}{(ax^{2} + bx + c)^{3}})a + 0 + 2(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})b^{2} + 0\\=&\frac{-48a^{3}x^{3}}{(ax^{2} + bx + c)^{4}} - \frac{72a^{2}bx^{2}}{(ax^{2} + bx + c)^{4}} + \frac{24a^{2}x}{(ax^{2} + bx + c)^{3}} - \frac{36ab^{2}x}{(ax^{2} + bx + c)^{4}} + \frac{12ab}{(ax^{2} + bx + c)^{3}} - \frac{6b^{3}}{(ax^{2} + bx + c)^{4}}\\\\ &\color{blue}{The\ 4th\ derivative\ of\ function:} \\&\frac{d\left( \frac{-48a^{3}x^{3}}{(ax^{2} + bx + c)^{4}} - \frac{72a^{2}bx^{2}}{(ax^{2} + bx + c)^{4}} + \frac{24a^{2}x}{(ax^{2} + bx + c)^{3}} - \frac{36ab^{2}x}{(ax^{2} + bx + c)^{4}} + \frac{12ab}{(ax^{2} + bx + c)^{3}} - \frac{6b^{3}}{(ax^{2} + bx + c)^{4}}\right)}{dx}\\=&-48(\frac{-4(a*2x + b + 0)}{(ax^{2} + bx + c)^{5}})a^{3}x^{3} - \frac{48a^{3}*3x^{2}}{(ax^{2} + bx + c)^{4}} - 72(\frac{-4(a*2x + b + 0)}{(ax^{2} + bx + c)^{5}})a^{2}bx^{2} - \frac{72a^{2}b*2x}{(ax^{2} + bx + c)^{4}} + 24(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})a^{2}x + \frac{24a^{2}}{(ax^{2} + bx + c)^{3}} - 36(\frac{-4(a*2x + b + 0)}{(ax^{2} + bx + c)^{5}})ab^{2}x - \frac{36ab^{2}}{(ax^{2} + bx + c)^{4}} + 12(\frac{-3(a*2x + b + 0)}{(ax^{2} + bx + c)^{4}})ab + 0 - 6(\frac{-4(a*2x + b + 0)}{(ax^{2} + bx + c)^{5}})b^{3} + 0\\=&\frac{384a^{4}x^{4}}{(ax^{2} + bx + c)^{5}} + \frac{768a^{3}bx^{3}}{(ax^{2} + bx + c)^{5}} - \frac{288a^{3}x^{2}}{(ax^{2} + bx + c)^{4}} + \frac{576a^{2}b^{2}x^{2}}{(ax^{2} + bx + c)^{5}} - \frac{288a^{2}bx}{(ax^{2} + bx + c)^{4}} + \frac{192ab^{3}x}{(ax^{2} + bx + c)^{5}} - \frac{72ab^{2}}{(ax^{2} + bx + c)^{4}} + \frac{24a^{2}}{(ax^{2} + bx + c)^{3}} + \frac{24b^{4}}{(ax^{2} + bx + c)^{5}}\\ \end{split}\end{equation} \]

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