Differentiate the following functions with respect to x -a x2-b x-cp x2-a x

D/dx(ax^2+bx+c/px^2+qx+r ) =(px^2+qx+r)d/dx(ax^2+bx+c) (ax^2+bx+c)d/dx(px^2+qx+r)/(px^2+qx+r)^2 =(px^2+qx+r)(2ax+b) (ax^2+bx+c)(2px+q)/(px^2+qx+r)^2 =2apx^3+2aq ...

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Byju's Answer

Standard XII

Mathematics

Quotient Rule of Differentiation

Differentiate...

Question

Differentiate the following functions with respect to x :

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

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Solution

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

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

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

$= \frac{2 a p x^{3} + 2 a q x^{2} + 2 a x r + b p x^{2} + b q x + b r - (2 a p x^{3} + 2 p b x^{2} + 2 p c x + q a x^{2} + b q x + c q)}{(p x^{2} + q x + r)^{2}}$

$= \frac{2 a p x^{3} - 2 a p x^{2} + 2 a q x^{2} + b p x^{2} - 2 q a x^{2} + 2 a r x + b q x - 2 p c x - b q x + b r - c q}{(p x^{2} + q x + r)^{2}}$

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

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

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