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