derivative h(w)=(w^2-1)/(w^2+1)

解

導関数

h (w) = \frac{w ^{2} - 1}{w ^{2} + 1}

\frac{4 w}{( w ^{2} + 1 ) ^{2}}

解答ステップ

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

商の法則を適用:

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

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

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

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

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

拡張

2 w (w^{2} + 1) - 2 w (w^{2} - 1) : 4 w

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