derivative f(x)=(4(x^2-1))/((4x^3-2x)^{2/3)}

解

導関数

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

\frac{4 ( 16 x ^{2} - 4 )}{3 ( 4 x ^{3} - 2 x ) ^{\frac{5}{3}}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

\frac{d}{d x} (x^{2} - 1) = 2 x

\frac{d}{d x} ((4 x^{3} - 2 x)^{\frac{2}{3}}) = \frac{2 ( 12 x ^{2} - 2 )}{3 ( 4 x ^{3} - 2 x ) ^{\frac{1}{3}}}

= 4 \cdot \frac{2 x ( 4 x ^{3} - 2 x ) ^{\frac{2}{3}} - \frac{2 ( 12 x ^{2} - 2 )}{3 ( 4 x ^{3} - 2 x ) ^{\frac{1}{3}}} ( x ^{2} - 1 )}{( ( 4 x ^{3} - 2 x ) ^{\frac{2}{3}} ) ^{2}}

簡素化

4 \cdot \frac{2 x ( 4 x ^{3} - 2 x ) ^{\frac{2}{3}} - \frac{2 ( 12 x ^{2} - 2 )}{3 ( 4 x ^{3} - 2 x ) ^{\frac{1}{3}}} ( x ^{2} - 1 )}{( ( 4 x ^{3} - 2 x ) ^{\frac{2}{3}} ) ^{2}} : \frac{4 ( 16 x ^{2} - 4 )}{3 ( 4 x ^{3} - 2 x ) ^{\frac{5}{3}}}

= \frac{4 ( 16 x ^{2} - 4 )}{3 ( 4 x ^{3} - 2 x ) ^{\frac{5}{3}}}