derivative of x/(\sqrt[3]{x^2+4})

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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