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

解

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

\frac{4 x ( x ^{2} + 1 ) ( - 3 x ^{4} - 4 x ^{2} + 1 )}{( x ^{4} + 1 ) ^{5}}

解答ステップ

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

商の法則を適用:

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

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

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

\frac{d}{d x} ((x^{4} + 1)^{4}) = 16 x^{3} (x^{4} + 1)^{3}

= \frac{4 x ( x ^{2} + 1 ) ( x ^{4} + 1 ) ^{4} - 16 x ^{3} ( x ^{4} + 1 ) ^{3} ( x ^{2} + 1 ) ^{2}}{( ( x ^{4} + 1 ) ^{4} ) ^{2}}

簡素化

\frac{4 x ( x ^{2} + 1 ) ( x ^{4} + 1 ) ^{4} - 16 x ^{3} ( x ^{4} + 1 ) ^{3} ( x ^{2} + 1 ) ^{2}}{( ( x ^{4} + 1 ) ^{4} ) ^{2}} : \frac{4 x ( x ^{2} + 1 ) ( - 3 x ^{4} - 4 x ^{2} + 1 )}{( x ^{4} + 1 ) ^{5}}

= \frac{4 x ( x ^{2} + 1 ) ( - 3 x ^{4} - 4 x ^{2} + 1 )}{( x ^{4} + 1 ) ^{5}}