derivative (2.25r)/((r+0.5)^2)

解

導関数

\frac{2.25 r}{( r + 0.5 ) ^{2}}

\frac{2.25 ( 0.5 - r )}{( r + 0.5 ) ^{3}}

解答ステップ

\frac{d}{d r} (\frac{2.25 r}{( r + 0.5 ) ^{2}})

定数を除去:

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

= 2.25 \frac{d}{d r} (\frac{r}{( r + 0.5 ) ^{2}})

商の法則を適用:

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

= 2.25 \cdot \frac{\frac{d r}{d r} ( r + 0.5 ) ^{2} - \frac{d}{d r} ( ( r + 0.5 ) ^{2} ) r}{( ( r + 0.5 ) ^{2} ) ^{2}}

\frac{d r}{d r} = 1

\frac{d}{d r} ((r + 0.5)^{2}) = 2 (r + 0.5)

= 2.25 \cdot \frac{1 \cdot ( r + 0.5 ) ^{2} - 2 ( r + 0.5 ) r}{( ( r + 0.5 ) ^{2} ) ^{2}}

簡素化

2.25 \cdot \frac{1 \cdot ( r + 0.5 ) ^{2} - 2 ( r + 0.5 ) r}{( ( r + 0.5 ) ^{2} ) ^{2}} : \frac{2.25 ( 0.5 - r )}{( r + 0.5 ) ^{3}}

= \frac{2.25 ( 0.5 - r )}{( r + 0.5 ) ^{3}}