derivative of ((10x^2-7x-12x)/((10z+8)(2x+3)))

解

\frac{d}{d x} (\frac{( 10 x ^{2} - 7 x - 12 ) x}{( 10 z ( x ) + 8 ) ( 2 x + 3 )})

\frac{- 200 x ^{4} \frac{d z}{d x} + 400 x ^{3} z + 320 x ^{3} - 160 x ^{3} \frac{d z}{d x} + 760 x ^{2} z + 608 x ^{2} + 450 x ^{2} \frac{d z}{d x} - 420 x z - 336 x + 360 x \frac{d z}{d x} - 360 z - 288}{( 10 z + 8 ) ^{2} ( 2 x + 3 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{( 10 x ^{2} - 7 x - 12 ) x}{( 10 z + 8 ) ( 2 x + 3 )})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( ( 10 x ^{2} - 7 x - 12 ) x ) ( 10 z + 8 ) ( 2 x + 3 ) - \frac{d}{d x} ( ( 10 z + 8 ) ( 2 x + 3 ) ) ( 10 x ^{2} - 7 x - 12 ) x}{( ( 10 z + 8 ) ( 2 x + 3 ) ) ^{2}}

\frac{d}{d x} ((10 x^{2} - 7 x - 12) x) = 30 x^{2} - 14 x - 12

\frac{d}{d x} ((10 z + 8) (2 x + 3)) = 10 \frac{d z}{d x} (2 x + 3) + 2 (10 z + 8)

= \frac{( 30 x ^{2} - 14 x - 12 ) ( 10 z + 8 ) ( 2 x + 3 ) - ( 10 \frac{d z}{d x} ( 2 x + 3 ) + 2 ( 10 z + 8 ) ) ( 10 x ^{2} - 7 x - 12 ) x}{( ( 10 z + 8 ) ( 2 x + 3 ) ) ^{2}}

簡素化

\frac{( 30 x ^{2} - 14 x - 12 ) ( 10 z + 8 ) ( 2 x + 3 ) - ( 10 \frac{d z}{d x} ( 2 x + 3 ) + 2 ( 10 z + 8 ) ) ( 10 x ^{2} - 7 x - 12 ) x}{( ( 10 z + 8 ) ( 2 x + 3 ) ) ^{2}} : \frac{- 200 x ^{4} \frac{d z}{d x} + 400 x ^{3} z + 320 x ^{3} - 160 x ^{3} \frac{d z}{d x} + 760 x ^{2} z + 608 x ^{2} + 450 x ^{2} \frac{d z}{d x} - 420 x z - 336 x + 360 x \frac{d z}{d x} - 360 z - 288}{( 10 z + 8 ) ^{2} ( 2 x + 3 ) ^{2}}

= \frac{- 200 x ^{4} \frac{d z}{d x} + 400 x ^{3} z + 320 x ^{3} - 160 x ^{3} \frac{d z}{d x} + 760 x ^{2} z + 608 x ^{2} + 450 x ^{2} \frac{d z}{d x} - 420 x z - 336 x + 360 x \frac{d z}{d x} - 360 z - 288}{( 10 z + 8 ) ^{2} ( 2 x + 3 ) ^{2}}