Question

Find the derivative of the following functions(it is to be understood that $a,b,c,d,p,q,r$ and $s$ are fixed non-zero constants and $m$ and $n$ are integers): $\frac{1}{a{x}^2+bx+c}$

Answer 1

Let $f\left(x\right)\frac{1}{a{x}^2+bx+c}$
Thus by quotient rule
$f^{\prime }\left(x\right)=\frac{(a{x}^2+bx+c)\frac{d}{dx}\left(1\right)-\frac{d}{dx}(a{x}^2+bx+c)}{{(a{x}^2+bx+c)}^2}$
$=\frac{(a{x}^2+bx+c)\left(0\right)-(2ax+b)}{{(a{x}^2+bx+c)}^2}=\frac{-(2ax+b)}{{(a{x}^2+bx+c)}^2}$