derivative of (x^2-9x+20/((x-2)(x+2)))

解

\frac{d}{d x} (\frac{x ^{2} - 9 x + 20}{( x - 2 ) ( x + 2 )})

\frac{9 x ^{2} - 48 x + 36}{( x - 2 ) ^{2} ( x + 2 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} - 9 x + 20}{( x - 2 ) ( 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 ^{2} - 9 x + 20 ) ( x - 2 ) ( x + 2 ) - \frac{d}{d x} ( ( x - 2 ) ( x + 2 ) ) ( x ^{2} - 9 x + 20 )}{( ( x - 2 ) ( x + 2 ) ) ^{2}}

\frac{d}{d x} (x^{2} - 9 x + 20) = 2 x - 9

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

= \frac{( 2 x - 9 ) ( x - 2 ) ( x + 2 ) - 2 x ( x ^{2} - 9 x + 20 )}{( ( x - 2 ) ( x + 2 ) ) ^{2}}

簡素化

\frac{( 2 x - 9 ) ( x - 2 ) ( x + 2 ) - 2 x ( x ^{2} - 9 x + 20 )}{( ( x - 2 ) ( x + 2 ) ) ^{2}} : \frac{9 x ^{2} - 48 x + 36}{( x - 2 ) ^{2} ( x + 2 ) ^{2}}

= \frac{9 x ^{2} - 48 x + 36}{( x - 2 ) ^{2} ( x + 2 ) ^{2}}