derivative f(x)=4e^{5x^2}(5-x)^7

解

導関数

f (x) = 4 e^{5 x^{2}} (5 - x)^{7}

4 (10 e^{5 x^{2}} x (5 - x)^{7} - 7 e^{5 x^{2}} (5 - x)^{6})

解答ステップ

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

定数を除去:

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

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

積の法則を適用:

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

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

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

\frac{d}{d x} (e^{5 x^{2}}) = e^{5 x^{2}} \cdot 10 x

\frac{d}{d x} ((5 - x)^{7}) = - 7 (5 - x)^{6}

= 4 (e^{5 x^{2}} \cdot 10 x (5 - x)^{7} + (- 7 (5 - x)^{6}) e^{5 x^{2}})

簡素化

= 4 (10 e^{5 x^{2}} x (5 - x)^{7} - 7 e^{5 x^{2}} (5 - x)^{6})