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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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