derivative f(x)=(1+16x^2)/((4x)^3)

解

導関数

f (x) = \frac{1 + 16 x ^{2}}{( 4 x ) ^{3}}

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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