derivative of-((2x^2+1)/((1-x^2)^{5/2)})

解

\frac{d}{d x} (- \frac{( 2 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{5}{2}}})

- \frac{3 x ( 2 x ^{2} + 3 )}{( 1 - x ^{2} ) ^{\frac{7}{2}}}

解答ステップ

\frac{d}{d x} (- \frac{( 2 x ^{2} + 1 )}{( 1 - x ^{2} ) ^{\frac{5}{2}}})

定数を除去:

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

= - \frac{d}{d x} (\frac{2 x ^{2} + 1}{( 1 - x ^{2} ) ^{\frac{5}{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 ^{2} + 1 ) ( 1 - x ^{2} ) ^{\frac{5}{2}} - \frac{d}{d x} ( ( 1 - x ^{2} ) ^{\frac{5}{2}} ) ( 2 x ^{2} + 1 )}{( ( 1 - x ^{2} ) ^{\frac{5}{2}} ) ^{2}}

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

\frac{d}{d x} ((1 - x^{2})^{\frac{5}{2}}) = - 5 x (1 - x^{2})^{\frac{3}{2}}

= - \frac{4 x ( 1 - x ^{2} ) ^{\frac{5}{2}} - ( - 5 x ( 1 - x ^{2} ) ^{\frac{3}{2}} ) ( 2 x ^{2} + 1 )}{( ( 1 - x ^{2} ) ^{\frac{5}{2}} ) ^{2}}

簡素化

- \frac{4 x ( 1 - x ^{2} ) ^{\frac{5}{2}} - ( - 5 x ( 1 - x ^{2} ) ^{\frac{3}{2}} ) ( 2 x ^{2} + 1 )}{( ( 1 - x ^{2} ) ^{\frac{5}{2}} ) ^{2}} : - \frac{3 x ( 2 x ^{2} + 3 )}{( 1 - x ^{2} ) ^{\frac{7}{2}}}

= - \frac{3 x ( 2 x ^{2} + 3 )}{( 1 - x ^{2} ) ^{\frac{7}{2}}}