derivative ((1+16t^2))/((4t)^3)

解

導関数

\frac{( 1 + 16 t ^{2} )}{( 4 t ) ^{3}}

\frac{- 16 t ^{2} - 3}{64 t ^{4}}

解答ステップ

\frac{d}{d t} (\frac{1 + 16 t ^{2}}{( 4 t ) ^{3}})

商の法則を適用:

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

= \frac{\frac{d}{d t} ( 1 + 16 t ^{2} ) ( 4 t ) ^{3} - \frac{d}{d t} ( ( 4 t ) ^{3} ) ( 1 + 16 t ^{2} )}{( ( 4 t ) ^{3} ) ^{2}}

\frac{d}{d t} (1 + 16 t^{2}) = 32 t

\frac{d}{d t} ((4 t)^{3}) = 192 t^{2}

= \frac{32 t ( 4 t ) ^{3} - 192 t ^{2} ( 1 + 16 t ^{2} )}{( ( 4 t ) ^{3} ) ^{2}}

簡素化

\frac{32 t ( 4 t ) ^{3} - 192 t ^{2} ( 1 + 16 t ^{2} )}{( ( 4 t ) ^{3} ) ^{2}} : \frac{- 16 t ^{2} - 3}{64 t ^{4}}

= \frac{- 16 t ^{2} - 3}{64 t ^{4}}