derivative of (4x/(2x^2+1))

Lösung

\frac{d}{d x} (\frac{4 x}{2 x ^{2} + 1})

\frac{4 ( - 2 x ^{2} + 1 )}{( 2 x ^{2} + 1 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{4 x}{2 x ^{2} + 1})

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d}{d x} (\frac{x}{2 x ^{2} + 1})

Wende die Quotientenregel an:

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

= 4 \cdot \frac{\frac{d x}{d x} ( 2 x ^{2} + 1 ) - \frac{d}{d x} ( 2 x ^{2} + 1 ) x}{( 2 x ^{2} + 1 ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} (2 x^{2} + 1) = 4 x

= 4 \cdot \frac{1 \cdot ( 2 x ^{2} + 1 ) - 4 xx}{( 2 x ^{2} + 1 ) ^{2}}

Vereinfache

4 \cdot \frac{1 \cdot ( 2 x ^{2} + 1 ) - 4 xx}{( 2 x ^{2} + 1 ) ^{2}} : \frac{4 ( - 2 x ^{2} + 1 )}{( 2 x ^{2} + 1 ) ^{2}}

= \frac{4 ( - 2 x ^{2} + 1 )}{( 2 x ^{2} + 1 ) ^{2}}

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