Find the derivative of px2-qx-rax-b where p-q- r- a- b are fixed non-zero constants-

Let f(x) = px^2+qx+rax+b Differentiating with respect to x ⇒ f' (x) = [ (ax +b)ddx(px^2 +qx +r) (px^2 +qx +r) ddx (ax +b)](ax +b)^2 ⇒ f' (x) = (2px + q)(ax+b) ...

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Byju's Answer

Standard XII

Physics

Basic Differentiation Rule

Find the deri...

Question

Find the derivative of $\frac{p x^{2} + q x + r}{a x + b}$ where $p, q, r, a, b$ are fixed non-zero constants.

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Solution

Let $f (x) = \frac{p x^{2} + q x + r}{a x + b}$
Differentiating with respect to $x$
$\Rightarrow f^{'} (x) = \frac{[(a x + b) \frac{d}{d x} (p x^{2} + q x + r) - (p x^{2} + q x + r) \frac{d}{d x} (a x + b)]}{(a x + b)^{2}}$
$\Rightarrow f^{'} (x) = \frac{(2 p x + q) (a x + b) - a (p x^{2} + q x + r)}{(a x + b)^{2}}$
$\Rightarrow f^{'} (x) = \frac{2 a p x^{2} + 2 b p x + q a x + b q - a p x^{2} - q a x - a r}{(a x + b)^{2}}$
$∴ f^{'} (x) = \frac{a p x^{2} + 2 b p x + b q - a r}{(a x + b)^{2}}$

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