derivative of (7x(2x-20^3)/(4x+9))

解

\frac{d}{d x} (\frac{7 x ( 2 x - 20 ) ^{3}}{4 x + 9})

\frac{7 ( 9 ( 2 x - 20 ) ^{3} + 24 x ^{2} ( 2 x - 20 ) ^{2} + 54 x ( 2 x - 20 ) ^{2} )}{( 4 x + 9 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{7 x ( 2 x - 20 ) ^{3}}{4 x + 9})

定数を除去:

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

= 7 \frac{d}{d x} (\frac{x ( 2 x - 20 ) ^{3}}{4 x + 9})

商の法則を適用:

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

= 7 \cdot \frac{\frac{d}{d x} ( x ( 2 x - 20 ) ^{3} ) ( 4 x + 9 ) - \frac{d}{d x} ( 4 x + 9 ) x ( 2 x - 20 ) ^{3}}{( 4 x + 9 ) ^{2}}

\frac{d}{d x} (x (2 x - 20)^{3}) = (2 x - 20)^{3} + 6 x (2 x - 20)^{2}

\frac{d}{d x} (4 x + 9) = 4

= 7 \cdot \frac{( ( 2 x - 20 ) ^{3} + 6 x ( 2 x - 20 ) ^{2} ) ( 4 x + 9 ) - 4 x ( 2 x - 20 ) ^{3}}{( 4 x + 9 ) ^{2}}

簡素化

7 \cdot \frac{( ( 2 x - 20 ) ^{3} + 6 x ( 2 x - 20 ) ^{2} ) ( 4 x + 9 ) - 4 x ( 2 x - 20 ) ^{3}}{( 4 x + 9 ) ^{2}} : \frac{7 ( 9 ( 2 x - 20 ) ^{3} + 24 x ^{2} ( 2 x - 20 ) ^{2} + 54 x ( 2 x - 20 ) ^{2} )}{( 4 x + 9 ) ^{2}}

= \frac{7 ( 9 ( 2 x - 20 ) ^{3} + 24 x ^{2} ( 2 x - 20 ) ^{2} + 54 x ( 2 x - 20 ) ^{2} )}{( 4 x + 9 ) ^{2}}