derivative of-(x-3^2(1-x)^2)

解

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

- 4 x^{3} + 24 x^{2} - 44 x + 24

解答ステップ

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

定数を除去:

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

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

積の法則を適用:

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

f = (x - 3)^{2}, g = (1 - x)^{2}

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

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

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

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

簡素化

- (2 (x - 3) (1 - x)^{2} + (- 2 (1 - x)) (x - 3)^{2}) : - 4 x^{3} + 24 x^{2} - 44 x + 24

= - 4 x^{3} + 24 x^{2} - 44 x + 24