derivative of ((x^3+2x^3)/(sqrt(2x-4)))

解

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

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

解答ステップ

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

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

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

\frac{d}{d x} (2 x - 4) = \frac{1}{2 x - 4}

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

簡素化

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

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