derivative y=(5x-3)^4(2x-7)^{-3}

解

導関数

y = (5 x - 3)^{4} (2 x - 7)^{- 3}

\frac{20 ( 5 x - 3 ) ^{3}}{( 2 x - 7 ) ^{3}} - \frac{6 ( 5 x - 3 ) ^{4}}{( 2 x - 7 ) ^{4}}

解答ステップ

\frac{d}{d x} ((5 x - 3)^{4} (2 x - 7)^{- 3})

積の法則を適用:

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

f = (5 x - 3)^{4}, g = (2 x - 7)^{- 3}

= \frac{d}{d x} ((5 x - 3)^{4}) (2 x - 7)^{- 3} + \frac{d}{d x} ((2 x - 7)^{- 3}) (5 x - 3)^{4}

\frac{d}{d x} ((5 x - 3)^{4}) = 20 (5 x - 3)^{3}

\frac{d}{d x} ((2 x - 7)^{- 3}) = - \frac{6}{( 2 x - 7 ) ^{4}}

= 20 (5 x - 3)^{3} (2 x - 7)^{- 3} + (- \frac{6}{( 2 x - 7 ) ^{4}}) (5 x - 3)^{4}

簡素化

20 (5 x - 3)^{3} (2 x - 7)^{- 3} + (- \frac{6}{( 2 x - 7 ) ^{4}}) (5 x - 3)^{4} : \frac{20 ( 5 x - 3 ) ^{3}}{( 2 x - 7 ) ^{3}} - \frac{6 ( 5 x - 3 ) ^{4}}{( 2 x - 7 ) ^{4}}

= \frac{20 ( 5 x - 3 ) ^{3}}{( 2 x - 7 ) ^{3}} - \frac{6 ( 5 x - 3 ) ^{4}}{( 2 x - 7 ) ^{4}}