derivative (1+5x^2)(x-x^2)

解

導関数

(1 + 5 x^{2}) (x - x^{2})

- 20 x^{3} + 15 x^{2} - 2 x + 1

解答ステップ

\frac{d}{d x} ((1 + 5 x^{2}) (x - x^{2}))

積の法則を適用:

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

f = 1 + 5 x^{2}, g = x - x^{2}

= \frac{d}{d x} (1 + 5 x^{2}) (x - x^{2}) + \frac{d}{d x} (x - x^{2}) (1 + 5 x^{2})

\frac{d}{d x} (1 + 5 x^{2}) = 10 x

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

= 10 x (x - x^{2}) + (1 - 2 x) (1 + 5 x^{2})

簡素化

10 x (x - x^{2}) + (1 - 2 x) (1 + 5 x^{2}) : - 20 x^{3} + 15 x^{2} - 2 x + 1

= - 20 x^{3} + 15 x^{2} - 2 x + 1