Differentiate the following functions with respect to x :

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

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Solution

Using quotient rule, we have

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

$= \frac{(a x^{2} + b x + c) \frac{d}{d x} (1) - 1 \times \frac{d}{d x} (a x^{2} + b x + c)}{(a x^{2} + b x + c)^{2}} = \frac{- 2 (a x + b)}{(a x^{2} + b x + c)^{2}}$

$∴ \frac{d}{d x} \frac{1}{a x^{2} + b x + c} = \frac{- (2 a x + b)}{(a x^{2} + b x + c)}$

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