derivative of (2x^2-1^3*(1-x^2)^{-3})

解

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

\frac{12 x ( 2 x ^{2} - 1 ) ^{2}}{( 1 - x ^{2} ) ^{3}} + \frac{6 x ( 2 x ^{2} - 1 ) ^{3}}{( 1 - x ^{2} ) ^{4}}

解答ステップ

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

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = (2 x^{2} - 1)^{3}, g = (1 - x^{2})^{- 3}

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

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

\frac{d}{d x} ((1 - x^{2})^{- 3}) = \frac{6 x}{( 1 - x ^{2} ) ^{4}}

= 12 x (2 x^{2} - 1)^{2} (1 - x^{2})^{- 3} + \frac{6 x}{( 1 - x ^{2} ) ^{4}} (2 x^{2} - 1)^{3}

簡素化

12 x (2 x^{2} - 1)^{2} (1 - x^{2})^{- 3} + \frac{6 x}{( 1 - x ^{2} ) ^{4}} (2 x^{2} - 1)^{3} : \frac{12 x ( 2 x ^{2} - 1 ) ^{2}}{( 1 - x ^{2} ) ^{3}} + \frac{6 x ( 2 x ^{2} - 1 ) ^{3}}{( 1 - x ^{2} ) ^{4}}

= \frac{12 x ( 2 x ^{2} - 1 ) ^{2}}{( 1 - x ^{2} ) ^{3}} + \frac{6 x ( 2 x ^{2} - 1 ) ^{3}}{( 1 - x ^{2} ) ^{4}}