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

解

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

\frac{8 x ^{3}}{( x ^{4} + 1 ) ^{2}}

解答ステップ

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

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

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

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

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

拡張

4 x^{3} (x^{4} + 1) - 4 x^{3} (x^{4} - 1) : 8 x^{3}

= \frac{8 x ^{3}}{( x ^{4} + 1 ) ^{2}}