Find the derivative of the following functions
Question:

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

Solution:

Let $f(x)=\frac{p x^{2}+q x+r}{a x+b}$

By quotient rule,

$f^{\prime}(x)=\frac{(a x+b) \frac{d}{d x}\left(p x^{2}+q x+r\right)-\left(p x^{2}+q x+r\right) \frac{d}{d x}(a x+b)}{(a x+b)^{2}}$

$=\frac{(a x+b)(2 p x+q)-\left(p x^{2}+q x+r\right)(a)}{(a x+b)^{2}}$

$=\frac{2 a p x^{2}+a q x+2 b p x+b q-a p x^{2}-a q x-a r}{(a x+b)^{2}}$

$=\frac{a p x^{2}+2 b p x+b q-a r}{(a x+b)^{2}}$

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