d/(ds)((6s^3+28s^2+38s+15)/((s+1)^2))

解

\frac{d}{d s} (\frac{6 s ^{3} + 28 s ^{2} + 38 s + 15}{( s + 1 ) ^{2}})

\frac{2 ( 3 s ^{3} + 9 s ^{2} + 9 s + 4 )}{( s + 1 ) ^{3}}

解答ステップ

\frac{d}{d s} (\frac{6 s ^{3} + 28 s ^{2} + 38 s + 15}{( s + 1 ) ^{2}})

商の法則を適用:

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

= \frac{\frac{d}{d s} ( 6 s ^{3} + 28 s ^{2} + 38 s + 15 ) ( s + 1 ) ^{2} - \frac{d}{d s} ( ( s + 1 ) ^{2} ) ( 6 s ^{3} + 28 s ^{2} + 38 s + 15 )}{( ( s + 1 ) ^{2} ) ^{2}}

\frac{d}{d s} (6 s^{3} + 28 s^{2} + 38 s + 15) = 18 s^{2} + 56 s + 38

\frac{d}{d s} ((s + 1)^{2}) = 2 (s + 1)

= \frac{( 18 s ^{2} + 56 s + 38 ) ( s + 1 ) ^{2} - 2 ( s + 1 ) ( 6 s ^{3} + 28 s ^{2} + 38 s + 15 )}{( ( s + 1 ) ^{2} ) ^{2}}

簡素化

\frac{( 18 s ^{2} + 56 s + 38 ) ( s + 1 ) ^{2} - 2 ( s + 1 ) ( 6 s ^{3} + 28 s ^{2} + 38 s + 15 )}{( ( s + 1 ) ^{2} ) ^{2}} : \frac{2 ( 3 s ^{3} + 9 s ^{2} + 9 s + 4 )}{( s + 1 ) ^{3}}

= \frac{2 ( 3 s ^{3} + 9 s ^{2} + 9 s + 4 )}{( s + 1 ) ^{3}}