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

解

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

\frac{2 ( - 15 x ^{4} - 66 x ^{3} + 2 x + 4 )}{( 3 x ^{3} + 2 ) ^{5}}

解答ステップ

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

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

\frac{d}{d x} ((3 x^{3} + 2)^{4}) = 36 x^{2} (3 x^{3} + 2)^{3}

= \frac{( 2 x + 4 ) ( 3 x ^{3} + 2 ) ^{4} - 36 x ^{2} ( 3 x ^{3} + 2 ) ^{3} ( x ^{2} + 4 x )}{( ( 3 x ^{3} + 2 ) ^{4} ) ^{2}}

簡素化

\frac{( 2 x + 4 ) ( 3 x ^{3} + 2 ) ^{4} - 36 x ^{2} ( 3 x ^{3} + 2 ) ^{3} ( x ^{2} + 4 x )}{( ( 3 x ^{3} + 2 ) ^{4} ) ^{2}} : \frac{2 ( - 15 x ^{4} - 66 x ^{3} + 2 x + 4 )}{( 3 x ^{3} + 2 ) ^{5}}

= \frac{2 ( - 15 x ^{4} - 66 x ^{3} + 2 x + 4 )}{( 3 x ^{3} + 2 ) ^{5}}