derivative of (x^2-9x+14/((x-5)(x-10)))

解

\frac{d}{d x} (\frac{x ^{2} - 9 x + 14}{( x - 5 ) ( x - 10 )})

\frac{- 6 x ^{2} + 72 x - 240}{( x - 5 ) ^{2} ( x - 10 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} - 9 x + 14}{( x - 5 ) ( x - 10 )})

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 + 14 ) ( x - 5 ) ( x - 10 ) - \frac{d}{d x} ( ( x - 5 ) ( x - 10 ) ) ( x ^{2} - 9 x + 14 )}{( ( x - 5 ) ( x - 10 ) ) ^{2}}

\frac{d}{d x} (x^{2} - 9 x + 14) = 2 x - 9

\frac{d}{d x} ((x - 5) (x - 10)) = 2 x - 15

= \frac{( 2 x - 9 ) ( x - 5 ) ( x - 10 ) - ( 2 x - 15 ) ( x ^{2} - 9 x + 14 )}{( ( x - 5 ) ( x - 10 ) ) ^{2}}

簡素化

\frac{( 2 x - 9 ) ( x - 5 ) ( x - 10 ) - ( 2 x - 15 ) ( x ^{2} - 9 x + 14 )}{( ( x - 5 ) ( x - 10 ) ) ^{2}} : \frac{- 6 x ^{2} + 72 x - 240}{( x - 5 ) ^{2} ( x - 10 ) ^{2}}

= \frac{- 6 x ^{2} + 72 x - 240}{( x - 5 ) ^{2} ( x - 10 ) ^{2}}