derivative f(x)=(3x)/((1-x)^2)

解

導関数

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

\frac{3 ( 1 + x )}{( 1 - x ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{3 x}{( 1 - x ) ^{2}})

定数を除去:

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

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

商の法則を適用:

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

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

\frac{d x}{d x} = 1

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

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

簡素化

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

= \frac{3 ( 1 + x )}{( 1 - x ) ^{3}}