derivative of (x-4/(3x-x^2))

解

\frac{d}{d x} (\frac{x - 4}{3 x - x ^{2}})

\frac{x ^{2} - 8 x + 12}{( 3 x - x ^{2} ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x - 4}{3 x - 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 - 4 ) ( 3 x - x ^{2} ) - \frac{d}{d x} ( 3 x - x ^{2} ) ( x - 4 )}{( 3 x - x ^{2} ) ^{2}}

\frac{d}{d x} (x - 4) = 1

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

= \frac{1 \cdot ( 3 x - x ^{2} ) - ( 3 - 2 x ) ( x - 4 )}{( 3 x - x ^{2} ) ^{2}}

簡素化

\frac{1 \cdot ( 3 x - x ^{2} ) - ( 3 - 2 x ) ( x - 4 )}{( 3 x - x ^{2} ) ^{2}} : \frac{x ^{2} - 8 x + 12}{( 3 x - x ^{2} ) ^{2}}

= \frac{x ^{2} - 8 x + 12}{( 3 x - x ^{2} ) ^{2}}