d/(dp)(((10p^{0.25}+1))/(p^{0.25)})

解

\frac{d}{d p} (\frac{( 10 p ^{0.25} + 1 )}{p ^{0.25}})

- \frac{0.25}{p ^{1.25}}

解答ステップ

\frac{d}{d p} (\frac{( 10 p ^{0.25} + 1 )}{p ^{0.25}})

商の法則を適用:

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

= \frac{\frac{d}{d p} ( 10 p ^{0.25} + 1 ) p ^{0.25} - \frac{d}{d p} ( p ^{0.25} ) ( 10 p ^{0.25} + 1 )}{( p ^{0.25} ) ^{2}}

\frac{d}{d p} (10 p^{0.25} + 1) = \frac{2.5}{p ^{0.75}}

\frac{d}{d p} (p^{0.25}) = \frac{0.25}{p ^{0.75}}

= \frac{\frac{2.5}{p ^{0.75}} p ^{0.25} - \frac{0.25}{p ^{0.75}} ( 10 p ^{0.25} + 1 )}{( p ^{0.25} ) ^{2}}

簡素化

\frac{\frac{2.5}{p ^{0.75}} p ^{0.25} - \frac{0.25}{p ^{0.75}} ( 10 p ^{0.25} + 1 )}{( p ^{0.25} ) ^{2}} : - \frac{0.25}{p ^{1.25}}

= - \frac{0.25}{p ^{1.25}}