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

解

導関数

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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