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

解

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

\frac{2 x ^{3} - 7 x ^{2} + 4 x - 1}{( x - 2 ) ^{2}}

解答ステップ

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

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

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

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

簡素化

\frac{( 3 x ^{2} - 2 x ) ( x - 2 ) - 1 \cdot ( x ^{3} - x ^{2} + 1 )}{( x - 2 ) ^{2}} : \frac{2 x ^{3} - 7 x ^{2} + 4 x - 1}{( x - 2 ) ^{2}}

= \frac{2 x ^{3} - 7 x ^{2} + 4 x - 1}{( x - 2 ) ^{2}}