derivative of (4x^4-2x/(4x^4+2x))

Lösung

\frac{d}{d x} (\frac{4 x ^{4} - 2 x}{4 x ^{4} + 2 x})

\frac{12 x ^{2}}{( 2 x ^{3} + 1 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{4 x ^{4} - 2 x}{4 x ^{4} + 2 x})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( 4 x ^{4} - 2 x ) ( 4 x ^{4} + 2 x ) - \frac{d}{d x} ( 4 x ^{4} + 2 x ) ( 4 x ^{4} - 2 x )}{( 4 x ^{4} + 2 x ) ^{2}}

\frac{d}{d x} (4 x^{4} - 2 x) = 16 x^{3} - 2

\frac{d}{d x} (4 x^{4} + 2 x) = 16 x^{3} + 2

= \frac{( 16 x ^{3} - 2 ) ( 4 x ^{4} + 2 x ) - ( 16 x ^{3} + 2 ) ( 4 x ^{4} - 2 x )}{( 4 x ^{4} + 2 x ) ^{2}}

Vereinfache

\frac{( 16 x ^{3} - 2 ) ( 4 x ^{4} + 2 x ) - ( 16 x ^{3} + 2 ) ( 4 x ^{4} - 2 x )}{( 4 x ^{4} + 2 x ) ^{2}} : \frac{12 x ^{2}}{( 2 x ^{3} + 1 ) ^{2}}

= \frac{12 x ^{2}}{( 2 x ^{3} + 1 ) ^{2}}