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Beliebt Rechnen >

derivative of (2(16-x^2)/((16+x^2)^2))

Lösung

\frac{d}{d x} (\frac{2 ( 16 - x ^{2} )}{( 16 + x ^{2} ) ^{2}})

Lösung

- \frac{4 x ( - x ^{2} + 48 )}{( x ^{2} + 16 ) ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{2 ( 16 - x ^{2} )}{( 16 + x ^{2} ) ^{2}})

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

= 2 \cdot \frac{\frac{d}{d x} ( 16 - x ^{2} ) ( 16 + x ^{2} ) ^{2} - \frac{d}{d x} ( ( 16 + x ^{2} ) ^{2} ) ( 16 - x ^{2} )}{( ( 16 + x ^{2} ) ^{2} ) ^{2}}

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

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

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

Vereinfache

2 \cdot \frac{( - 2 x ) ( 16 + x ^{2} ) ^{2} - 4 x ( 16 + x ^{2} ) ( 16 - x ^{2} )}{( ( 16 + x ^{2} ) ^{2} ) ^{2}} : - \frac{4 x ( - x ^{2} + 48 )}{( x ^{2} + 16 ) ^{3}}

= - \frac{4 x ( - x ^{2} + 48 )}{( x ^{2} + 16 ) ^{3}}

Graph

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