derivative (410t)/((t^2+5)^3)

解

導関数

\frac{410 t}{( t ^{2} + 5 ) ^{3}}

\frac{2050 ( - t ^{2} + 1 )}{( t ^{2} + 5 ) ^{4}}

解答ステップ

\frac{d}{d t} (\frac{410 t}{( t ^{2} + 5 ) ^{3}})

定数を除去:

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

= 410 \frac{d}{d t} (\frac{t}{( t ^{2} + 5 ) ^{3}})

商の法則を適用:

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

= 410 \cdot \frac{\frac{d t}{d t} ( t ^{2} + 5 ) ^{3} - \frac{d}{d t} ( ( t ^{2} + 5 ) ^{3} ) t}{( ( t ^{2} + 5 ) ^{3} ) ^{2}}

\frac{d t}{d t} = 1

\frac{d}{d t} ((t^{2} + 5)^{3}) = 6 t (t^{2} + 5)^{2}

= 410 \cdot \frac{1 \cdot ( t ^{2} + 5 ) ^{3} - 6 t ( t ^{2} + 5 ) ^{2} t}{( ( t ^{2} + 5 ) ^{3} ) ^{2}}

簡素化

410 \cdot \frac{1 \cdot ( t ^{2} + 5 ) ^{3} - 6 t ( t ^{2} + 5 ) ^{2} t}{( ( t ^{2} + 5 ) ^{3} ) ^{2}} : \frac{2050 ( - t ^{2} + 1 )}{( t ^{2} + 5 ) ^{4}}

= \frac{2050 ( - t ^{2} + 1 )}{( t ^{2} + 5 ) ^{4}}