exercise - differentiation but you factorize first ♡

worked example: Differentiate $f (x) = \frac{x + 3}{2 x ^{2} + 5 x - 3}$ by factorizing first

$f (x) = \frac{x + 3}{( x + 3 ) ( 2 x - 1 )}$
$f (x) = (2 x - 1)^{- 1}$

$f^{'} (x) = (- 1) (2 x - 1)^{- 2} \cdot (2)$
$f^{'} (x) = \frac{- 2}{( 2 x - 1 ) ^{2}}$

worked example: Differentiate $f (x) = \frac{x + 3}{2 x ^{2} + 5 x - 3}$

$f^{'} (x) = \frac{( 1 ) ( 2 x ^{2} + 5 x - 3 ) - ( x + 3 ) ( 4 x + 5 )}{( 2 x ^{2} + 5 x - 3 ) ^{2}}$

$f^{'} (x) = \frac{2 x ^{2} + 5 x - 3 - ( 4 x ^{2} + 5 x + 12 x + 15 )}{( 2 x ^{2} + 5 x - 3 ) ^{2}}$

$f^{'} (x) = \frac{- 2 x ^{2} - 12 x - 18}{( 2 x ^{2} + 5 x - 3 ) ^{2}}$

worked example: Differentiate $f (x) = \frac{x + 1}{2 x ^{2} - x - 3}$ by factorizing first

$f (x) = \frac{x + 1}{( x + 1 ) ( 2 x - 3 )}$

$f (x) = (2 x - 3)^{- 1}$

$f^{'} (x) = (- 1) (2 x - 3)^{- 2} \cdot (2)$

$f^{'} (x) = \frac{- 2}{( 2 x - 3 ) ^{2}}$

worked example: Differentiate $f (x) = \frac{x + 1}{2 x ^{2} - x - 3}$

$f^{'} (x) = \frac{( 1 ) ( 2 x ^{2} - x - 3 ) - ( x + 1 ) ( 4 x - 1 )}{( 2 x ^{2} - x - 3 ) ^{2}}$

$f^{'} (x) = \frac{2 x ^{2} - x - 3 - ( 4 x ^{2} - x + 4 x - 1 )}{( 2 x ^{2} - x - 3 ) ^{2}}$

$f^{'} (x) = \frac{- 2 x ^{2} - 4 x - 2}{( 2 x ^{2} - x - 3 ) ^{2}}$

worked example: Differentiate $f (x) = \frac{2 x + 1}{6 x ^{2} + x - 1}$ by factorizing first

$f (x) = \frac{2 x + 1}{( 3 x - 1 ) ( 2 x + 1 )}$

$f^{'} (x) = (3 x - 1)^{- 1}$

$f^{'} (x) = (- 1) (3 x - 1)^{- 2} \cdot (3)$

$f^{'} (x) = \frac{- 3}{( 3 x - 1 ) ^{2}}$

worked example: Differentiate $f (x) = \frac{2 x + 1}{6 x ^{2} + x - 1}$

$f^{'} (x) = \frac{( 2 ) ( 6 x ^{2} + x - 1 ) - ( 2 x + 1 ) ( 12 x + 1 )}{( 6 x ^{2} + x - 1 ) ^{2}}$

$f^{'} (x) = \frac{12 x ^{2} + 2 x - 2 - ( 24 x ^{2} + 2 x + 12 x + 1 )}{( 6 x ^{2} + x - 1 ) ^{2}}$

$f^{'} (x) = \frac{- 12 x ^{2} - 12 x - 3}{( 6 x ^{2} + x - 1 ) ^{2}}$

wiki ♡