derivative of (4x/(1+2x^2))

解

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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