derivative of (3x^2+5x+8/(\sqrt[3]{x)})

解

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

\frac{18 3 x x ^{\frac{5}{3}} + 15 3 x x ^{\frac{2}{3}} - 3 x ^{2} - 5 x - 8}{3 x ^{\frac{4}{3}}}

解答ステップ

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

商の法則を適用:

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

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

\frac{d}{d x} (3 x^{2} + 5 x + 8) = 6 x + 5

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

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

簡素化

\frac{( 6 x + 5 ) 3 x - \frac{1}{3 x ^{\frac{2}{3}}} ( 3 x ^{2} + 5 x + 8 )}{( 3 x ) ^{2}} : \frac{18 3 x x ^{\frac{5}{3}} + 15 3 x x ^{\frac{2}{3}} - 3 x ^{2} - 5 x - 8}{3 x ^{\frac{4}{3}}}

= \frac{18 3 x x ^{\frac{5}{3}} + 15 3 x x ^{\frac{2}{3}} - 3 x ^{2} - 5 x - 8}{3 x ^{\frac{4}{3}}}