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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

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

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