d/(dt)((3t^2)/(sqrt(1+4t^2+9t^4)))

解

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

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

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

= 3 \cdot \frac{\frac{d}{d t} ( t ^{2} ) 1 + 4 t ^{2} + 9 t ^{4} - \frac{d}{d t} ( 1 + 4 t ^{2} + 9 t ^{4} ) t ^{2}}{( 1 + 4 t ^{2} + 9 t ^{4} ) ^{2}}

\frac{d}{d t} (t^{2}) = 2 t

\frac{d}{d t} (1 + 4 t^{2} + 9 t^{4}) = \frac{2 t ( 9 t ^{2} + 2 )}{1 + 4 t ^{2} + 9 t ^{4}}

= 3 \cdot \frac{2 t 1 + 4 t ^{2} + 9 t ^{4} - \frac{2 t ( 9 t ^{2} + 2 )}{1 + 4 t ^{2} + 9 t ^{4}} t ^{2}}{( 1 + 4 t ^{2} + 9 t ^{4} ) ^{2}}

簡素化

3 \cdot \frac{2 t 1 + 4 t ^{2} + 9 t ^{4} - \frac{2 t ( 9 t ^{2} + 2 )}{1 + 4 t ^{2} + 9 t ^{4}} t ^{2}}{( 1 + 4 t ^{2} + 9 t ^{4} ) ^{2}} : \frac{3 ( 4 t ^{3} + 2 t )}{( 1 + 4 t ^{2} + 9 t ^{4} ) 1 + 4 t ^{2} + 9 t ^{4}}

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