Differentiate the following functions with respect to x :

$\frac{p x^{2} + q x + r}{a x + b}$

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Solution

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

Differentiating y w.r.t. x, we get

$\frac{d y}{d x} = \frac{(a x + b) \frac{d}{d x} (p x^{2} + q x + r) - (p x^{2} + q x + r) \frac{d}{d x} (a x + b)}{(a x + b)^{2}}$ (By quotient formula)

$\frac{(a x + b) (2 p x + q + 0) - (p x^{2} + q x + r) (a \times 1 + 0)}{(a x + b)^{2}}$

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

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

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

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Differentiate the following functions with respect to x :

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= \frac{p x^{2} + q x + r}{a x + b}

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