derivative of ((x+1(x-2))/(x^2-5x+1))

解

\frac{d}{d x} (\frac{( x + 1 ) ( x - 2 )}{x ^{2} - 5 x + 1})

\frac{- 4 x ^{2} + 6 x - 11}{( x ^{2} - 5 x + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{( x + 1 ) ( x - 2 )}{x ^{2} - 5 x + 1})

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 + 1 ) ( x - 2 ) ) ( x ^{2} - 5 x + 1 ) - \frac{d}{d x} ( x ^{2} - 5 x + 1 ) ( x + 1 ) ( x - 2 )}{( x ^{2} - 5 x + 1 ) ^{2}}

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

\frac{d}{d x} (x^{2} - 5 x + 1) = 2 x - 5

= \frac{( 2 x - 1 ) ( x ^{2} - 5 x + 1 ) - ( 2 x - 5 ) ( x + 1 ) ( x - 2 )}{( x ^{2} - 5 x + 1 ) ^{2}}

拡張

(2 x - 1) (x^{2} - 5 x + 1) - (2 x - 5) (x + 1) (x - 2) : - 4 x^{2} + 6 x - 11

= \frac{- 4 x ^{2} + 6 x - 11}{( x ^{2} - 5 x + 1 ) ^{2}}