derivative y=(x^2-8x+3)/(x^2+5)

解

導関数

y = \frac{x ^{2} - 8 x + 3}{x ^{2} + 5}

\frac{8 x ^{2} + 4 x - 40}{( x ^{2} + 5 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} - 8 x + 3}{x ^{2} + 5})

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 + 3 ) ( x ^{2} + 5 ) - \frac{d}{d x} ( x ^{2} + 5 ) ( x ^{2} - 8 x + 3 )}{( x ^{2} + 5 ) ^{2}}

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

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

= \frac{( 2 x - 8 ) ( x ^{2} + 5 ) - 2 x ( x ^{2} - 8 x + 3 )}{( x ^{2} + 5 ) ^{2}}

拡張

(2 x - 8) (x^{2} + 5) - 2 x (x^{2} - 8 x + 3) : 8 x^{2} + 4 x - 40

= \frac{8 x ^{2} + 4 x - 40}{( x ^{2} + 5 ) ^{2}}