derivative of (x^3-8/(x^2-4))

解

\frac{d}{d x} (\frac{x ^{3} - 8}{x ^{2} - 4})

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

解答ステップ

\frac{d}{d x} (\frac{x ^{3} - 8}{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 ^{3} - 8 ) ( x ^{2} - 4 ) - \frac{d}{d x} ( x ^{2} - 4 ) ( x ^{3} - 8 )}{( x ^{2} - 4 ) ^{2}}

\frac{d}{d x} (x^{3} - 8) = 3 x^{2}

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

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

簡素化

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

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