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

解

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

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

解答ステップ

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

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

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

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

拡張

2 x (x^{2} + 4) - 2 x (x^{2} - 4) : 16 x

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