derivative of (x^5+2x+6/(x^{1/3)})

解

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

\frac{14 x ^{5} + 4 x - 6}{3 x ^{\frac{4}{3}}}

解答ステップ

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

商の法則を適用:

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

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

\frac{d}{d x} (x^{5} + 2 x + 6) = 5 x^{4} + 2

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

= \frac{( 5 x ^{4} + 2 ) x ^{\frac{1}{3}} - \frac{1}{3 x ^{\frac{2}{3}}} ( x ^{5} + 2 x + 6 )}{( x ^{\frac{1}{3}} ) ^{2}}

簡素化

\frac{( 5 x ^{4} + 2 ) x ^{\frac{1}{3}} - \frac{1}{3 x ^{\frac{2}{3}}} ( x ^{5} + 2 x + 6 )}{( x ^{\frac{1}{3}} ) ^{2}} : \frac{14 x ^{5} + 4 x - 6}{3 x ^{\frac{4}{3}}}

= \frac{14 x ^{5} + 4 x - 6}{3 x ^{\frac{4}{3}}}