derivative of (2x^5/(5+3x^4+6x^3+2x+4))

解

\frac{d}{d x} (\frac{2 x ^{5}}{5 + 3 x ^{4} + 6 x ^{3} + 2 x + 4})

\frac{2 ( 3 x ^{8} + 12 x ^{7} + 8 x ^{5} + 45 x ^{4} )}{( 3 x ^{4} + 6 x ^{3} + 2 x + 9 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{2 x ^{5}}{5 + 3 x ^{4} + 6 x ^{3} + 2 x + 4})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

商の法則を適用:

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

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

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

\frac{d}{d x} (5 + 3 x^{4} + 6 x^{3} + 2 x + 4) = 12 x^{3} + 18 x^{2} + 2

= 2 \cdot \frac{5 x ^{4} ( 5 + 3 x ^{4} + 6 x ^{3} + 2 x + 4 ) - ( 12 x ^{3} + 18 x ^{2} + 2 ) x ^{5}}{( 5 + 3 x ^{4} + 6 x ^{3} + 2 x + 4 ) ^{2}}

簡素化

2 \cdot \frac{5 x ^{4} ( 5 + 3 x ^{4} + 6 x ^{3} + 2 x + 4 ) - ( 12 x ^{3} + 18 x ^{2} + 2 ) x ^{5}}{( 5 + 3 x ^{4} + 6 x ^{3} + 2 x + 4 ) ^{2}} : \frac{2 ( 3 x ^{8} + 12 x ^{7} + 8 x ^{5} + 45 x ^{4} )}{( 3 x ^{4} + 6 x ^{3} + 2 x + 9 ) ^{2}}

= \frac{2 ( 3 x ^{8} + 12 x ^{7} + 8 x ^{5} + 45 x ^{4} )}{( 3 x ^{4} + 6 x ^{3} + 2 x + 9 ) ^{2}}