derivative of (3x-1^4(7x^2-4)^{-3})

解

\frac{d}{d x} ((3 x - 1)^{4} (7 x^{2} - 4)^{- 3})

\frac{12 ( 3 x - 1 ) ^{3}}{( 7 x ^{2} - 4 ) ^{3}} - \frac{42 x ( 3 x - 1 ) ^{4}}{( 7 x ^{2} - 4 ) ^{4}}

解答ステップ

\frac{d}{d x} ((3 x - 1)^{4} (7 x^{2} - 4)^{- 3})

積の法則を適用:

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

f = (3 x - 1)^{4}, g = (7 x^{2} - 4)^{- 3}

= \frac{d}{d x} ((3 x - 1)^{4}) (7 x^{2} - 4)^{- 3} + \frac{d}{d x} ((7 x^{2} - 4)^{- 3}) (3 x - 1)^{4}

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

\frac{d}{d x} ((7 x^{2} - 4)^{- 3}) = - \frac{42 x}{( 7 x ^{2} - 4 ) ^{4}}

= 12 (3 x - 1)^{3} (7 x^{2} - 4)^{- 3} + (- \frac{42 x}{( 7 x ^{2} - 4 ) ^{4}}) (3 x - 1)^{4}

簡素化

12 (3 x - 1)^{3} (7 x^{2} - 4)^{- 3} + (- \frac{42 x}{( 7 x ^{2} - 4 ) ^{4}}) (3 x - 1)^{4} : \frac{12 ( 3 x - 1 ) ^{3}}{( 7 x ^{2} - 4 ) ^{3}} - \frac{42 x ( 3 x - 1 ) ^{4}}{( 7 x ^{2} - 4 ) ^{4}}

= \frac{12 ( 3 x - 1 ) ^{3}}{( 7 x ^{2} - 4 ) ^{3}} - \frac{42 x ( 3 x - 1 ) ^{4}}{( 7 x ^{2} - 4 ) ^{4}}