derivative (2+u^2)^5(5-9u^2)^8

解

導関数

(2 + u^{2})^{5} (5 - 9 u^{2})^{8}

10 u (2 + u^{2})^{4} (5 - 9 u^{2})^{8} - 144 u (5 - 9 u^{2})^{7} (2 + u^{2})^{5}

解答ステップ

\frac{d}{d u} ((2 + u^{2})^{5} (5 - 9 u^{2})^{8})

積の法則を適用:

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

f = (2 + u^{2})^{5}, g = (5 - 9 u^{2})^{8}

= \frac{d}{d u} ((2 + u^{2})^{5}) (5 - 9 u^{2})^{8} + \frac{d}{d u} ((5 - 9 u^{2})^{8}) (2 + u^{2})^{5}

\frac{d}{d u} ((2 + u^{2})^{5}) = 10 u (2 + u^{2})^{4}

\frac{d}{d u} ((5 - 9 u^{2})^{8}) = - 144 u (5 - 9 u^{2})^{7}

= 10 u (2 + u^{2})^{4} (5 - 9 u^{2})^{8} + (- 144 u (5 - 9 u^{2})^{7}) (2 + u^{2})^{5}

簡素化

= 10 u (2 + u^{2})^{4} (5 - 9 u^{2})^{8} - 144 u (5 - 9 u^{2})^{7} (2 + u^{2})^{5}