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): $\frac{1}{a x^{2}+b x+c}$
Solution:
Let $f(x)=\frac{1}{a x^{2}+b x+c}$
By quotient rule,
$f^{\prime}(x)=\frac{\left(a x^{2}+b x+c\right) \frac{d}{d x}(1)-\frac{d}{d x}\left(a x^{2}+b x+c\right)}{\left(a x^{2}+b x+c\right)^{2}}$
$=\frac{\left(a x^{2}+b x+c\right)(0)-(2 a x+b)}{\left(a x^{2}+b x+c\right)^{2}}$
$=\frac{-(2 a x+b)}{\left(a x^{2}+b x+c\right)^{2}}$