derivative of (3x/((x^2+1)^{3/5)})

解

\frac{d}{d x} (\frac{3 x}{( x ^{2} + 1 ) ^{\frac{3}{5}}})

\frac{3 ( - x ^{2} + 5 )}{5 ( x ^{2} + 1 ) ^{\frac{8}{5}}}

解答ステップ

\frac{d}{d x} (\frac{3 x}{( x ^{2} + 1 ) ^{\frac{3}{5}}})

定数を除去:

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

= 3 \frac{d}{d x} (\frac{x}{( x ^{2} + 1 ) ^{\frac{3}{5}}})

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} ( x ^{2} + 1 ) ^{\frac{3}{5}} - \frac{d}{d x} ( ( x ^{2} + 1 ) ^{\frac{3}{5}} ) x}{( ( x ^{2} + 1 ) ^{\frac{3}{5}} ) ^{2}}

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

\frac{d}{d x} ((x^{2} + 1)^{\frac{3}{5}}) = \frac{6 x}{5 ( x ^{2} + 1 ) ^{\frac{2}{5}}}

= 3 \cdot \frac{1 \cdot ( x ^{2} + 1 ) ^{\frac{3}{5}} - \frac{6 x}{5 ( x ^{2} + 1 ) ^{\frac{2}{5}}} x}{( ( x ^{2} + 1 ) ^{\frac{3}{5}} ) ^{2}}

簡素化

3 \cdot \frac{1 \cdot ( x ^{2} + 1 ) ^{\frac{3}{5}} - \frac{6 x}{5 ( x ^{2} + 1 ) ^{\frac{2}{5}}} x}{( ( x ^{2} + 1 ) ^{\frac{3}{5}} ) ^{2}} : \frac{3 ( - x ^{2} + 5 )}{5 ( x ^{2} + 1 ) ^{\frac{8}{5}}}

= \frac{3 ( - x ^{2} + 5 )}{5 ( x ^{2} + 1 ) ^{\frac{8}{5}}}