derivative of ((-2x^3-1)/((x^3-1)^2))

解

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

- \frac{6 x ^{2} ( - x ^{3} - 2 )}{( x ^{3} - 1 ) ^{3}}

解答ステップ

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

商の法則を適用:

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

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

\frac{d}{d x} (- 2 x^{3} - 1) = - 6 x^{2}

\frac{d}{d x} ((x^{3} - 1)^{2}) = 6 x^{2} (x^{3} - 1)

= \frac{( - 6 x ^{2} ) ( x ^{3} - 1 ) ^{2} - 6 x ^{2} ( x ^{3} - 1 ) ( - 2 x ^{3} - 1 )}{( ( x ^{3} - 1 ) ^{2} ) ^{2}}

簡素化

\frac{( - 6 x ^{2} ) ( x ^{3} - 1 ) ^{2} - 6 x ^{2} ( x ^{3} - 1 ) ( - 2 x ^{3} - 1 )}{( ( x ^{3} - 1 ) ^{2} ) ^{2}} : - \frac{6 x ^{2} ( - x ^{3} - 2 )}{( x ^{3} - 1 ) ^{3}}

= - \frac{6 x ^{2} ( - x ^{3} - 2 )}{( x ^{3} - 1 ) ^{3}}