derivative of 20x(1-x^3)

解

\frac{d}{d x} (20 x (1 - x)^{3})

20 ((1 - x)^{3} - 3 x (1 - x)^{2})

解答ステップ

\frac{d}{d x} (20 x (1 - x)^{3})

定数を除去:

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

= 20 \frac{d}{d x} (x (1 - x)^{3})

積の法則を適用:

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

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

= 20 (\frac{d x}{d x} (1 - x)^{3} + \frac{d}{d x} ((1 - x)^{3}) x)

\frac{d x}{d x} = 1

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

= 20 (1 \cdot (1 - x)^{3} + (- 3 (1 - x)^{2}) x)

簡素化

= 20 ((1 - x)^{3} - 3 x (1 - x)^{2})