d/(d{r)}((200{r}({r}+2))/((2+3{r))^2})

解

\frac{d}{d r} (\frac{200 r ( r + 2 )}{( 2 + 3 r ) ^{2}})

\frac{400 ( - r + 2 )}{( 2 + 3 r ) ^{3}}

解答ステップ

\frac{d}{d r} (\frac{200 r ( r + 2 )}{( 2 + 3 r ) ^{2}})

定数を除去:

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

= 200 \frac{d}{d r} (\frac{r ( r + 2 )}{( 2 + 3 r ) ^{2}})

商の法則を適用:

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

= 200 \cdot \frac{\frac{d}{d r} ( r ( r + 2 ) ) ( 2 + 3 r ) ^{2} - \frac{d}{d r} ( ( 2 + 3 r ) ^{2} ) r ( r + 2 )}{( ( 2 + 3 r ) ^{2} ) ^{2}}

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

\frac{d}{d r} ((2 + 3 r)^{2}) = 6 (2 + 3 r)

= 200 \cdot \frac{( 2 r + 2 ) ( 2 + 3 r ) ^{2} - 6 ( 2 + 3 r ) r ( r + 2 )}{( ( 2 + 3 r ) ^{2} ) ^{2}}

簡素化

200 \cdot \frac{( 2 r + 2 ) ( 2 + 3 r ) ^{2} - 6 ( 2 + 3 r ) r ( r + 2 )}{( ( 2 + 3 r ) ^{2} ) ^{2}} : \frac{400 ( - r + 2 )}{( 2 + 3 r ) ^{3}}

= \frac{400 ( - r + 2 )}{( 2 + 3 r ) ^{3}}