Let $f(x) = \large\frac{1}{ax^2+bx+c}$

By quotient rule,

$ f'(x) = \large\frac{(ax^2+bx+c)\large\frac{d}{dx}(1)-\large\frac{d}{dx}(ax^2+bx+c)}{(ax^2+bx+c)^2}$

$ = \large\frac{(ax^2+bx+c)(0)-(2ax+b)}{(ax^2+bx+c)^2}$

$ = \large\frac{-(2ax+b)}{(ax^2+bx+c)^2}$