practice quotient rule ♡

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

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

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

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

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

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

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

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

worked example: Differentiate $f (x) = \frac{4 e ^{2 x} - 2 e ^{x} + 1}{2 e ^{2 x}}$

$f^{'} (x) = \frac{( 8 e ^{2 x} - 2 e ^{x} ) ( 2 e ^{2 x} ) - ( 4 e ^{2 x} - 2 e ^{x} ) ( 4 e ^{2 x} )}{( 2 e ^{2 x} ) ^{2}}$

$f^{'} (x) = \frac{16 e ^{4 x} - 4 e ^{3 x} - 16 e ^{4 x} + 8 e ^{3 x}}{2 e ^{4 x}}$

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

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

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

wiki ♡