derivative of (x^3-2/((x-1)^2))

解

\frac{d}{d x} (\frac{x ^{3} - 2}{( x - 1 ) ^{2}})

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

解答ステップ

\frac{d}{d x} (\frac{x ^{3} - 2}{( x - 1 ) ^{2}})

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} - 2 ) ( x - 1 ) ^{2} - \frac{d}{d x} ( ( x - 1 ) ^{2} ) ( x ^{3} - 2 )}{( ( x - 1 ) ^{2} ) ^{2}}

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

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

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

簡素化

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

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