derivative (x-2)/((x+2)(x-3))

解

導関数

\frac{x - 2}{( x + 2 ) ( x - 3 )}

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

解答ステップ

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

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

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

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

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

簡素化

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

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