derivative of (4x^{2/3}/(x^2+1))

解

\frac{d}{d x} (\frac{4 x ^{\frac{2}{3}}}{x ^{2} + 1})

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

解答ステップ

\frac{d}{d x} (\frac{4 x ^{\frac{2}{3}}}{x ^{2} + 1})

定数を除去:

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

= 4 \frac{d}{d x} (\frac{x ^{\frac{2}{3}}}{x ^{2} + 1})

商の法則を適用:

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

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

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

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

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

簡素化

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

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