Find the derivative of ax-bpx2 -qx -r where p- q- r- a- b are fixed non-zero constants-

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

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Byju's Answer

Standard XII

Mathematics

Addition Rule of Differentiation

Find the deri...

Question

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

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Solution

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

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Similar questions

Q. 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.

Q. Find the derivative of the following functions(it is to be understood that

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and

s

are fixed non-zero constants and

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

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):