derivative f(x)=(2x^2)/(x^2-2x-8)

解

導関数

f (x) = \frac{2 x ^{2}}{x ^{2} - 2 x - 8}

\frac{2 ( - 2 x ^{2} - 16 x )}{( x ^{2} - 2 x - 8 ) ^{2}}

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

= 2 \cdot \frac{2 x ( x ^{2} - 2 x - 8 ) - ( 2 x - 2 ) x ^{2}}{( x ^{2} - 2 x - 8 ) ^{2}}

簡素化

2 \cdot \frac{2 x ( x ^{2} - 2 x - 8 ) - ( 2 x - 2 ) x ^{2}}{( x ^{2} - 2 x - 8 ) ^{2}} : \frac{2 ( - 2 x ^{2} - 16 x )}{( x ^{2} - 2 x - 8 ) ^{2}}

= \frac{2 ( - 2 x ^{2} - 16 x )}{( x ^{2} - 2 x - 8 ) ^{2}}