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): $\frac{1}{a x^{2} + b x + c}$

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Solution

Let $f (x) \frac{1}{a x^{2} + b x + c}$
Thus by quotient rule
$f^{'} (x) = \frac{(a x^{2} + b x + c) \frac{d}{d x} (1) - \frac{d}{d x} (a x^{2} + b x + c)}{{(a x^{2} + b x + c)}^{2}}$
$= \frac{(a x^{2} + b x + c) (0) - (2 a x + b)}{{(a x^{2} + b x + c)}^{2}} = \frac{- (2 a x + b)}{{(a x^{2} + b x + c)}^{2}}$

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