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

解

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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