derivative of (x^2-8x+15/(x^2+3x+2))

解

\frac{d}{d x} (\frac{x ^{2} - 8 x + 15}{x ^{2} + 3 x + 2})

\frac{11 x ^{2} - 26 x - 61}{( x ^{2} + 3 x + 2 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} - 8 x + 15}{x ^{2} + 3 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} - 8 x + 15 ) ( x ^{2} + 3 x + 2 ) - \frac{d}{d x} ( x ^{2} + 3 x + 2 ) ( x ^{2} - 8 x + 15 )}{( x ^{2} + 3 x + 2 ) ^{2}}

\frac{d}{d x} (x^{2} - 8 x + 15) = 2 x - 8

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

= \frac{( 2 x - 8 ) ( x ^{2} + 3 x + 2 ) - ( 2 x + 3 ) ( x ^{2} - 8 x + 15 )}{( x ^{2} + 3 x + 2 ) ^{2}}

拡張

(2 x - 8) (x^{2} + 3 x + 2) - (2 x + 3) (x^{2} - 8 x + 15) : 11 x^{2} - 26 x - 61

= \frac{11 x ^{2} - 26 x - 61}{( x ^{2} + 3 x + 2 ) ^{2}}