derivative of (2x+sqrt(x)(x^2-x))

解

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

\frac{x ^{\frac{3}{2}} - 3 x + 12 x ^{2} - 8 x + 4 x x}{2}

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

(2 + \frac{1}{2 x}) (x^{2} - x) + (2 x - 1) (2 x + x) : \frac{x ^{\frac{3}{2}} - 3 x + 12 x ^{2} - 8 x + 4 x x}{2}

= \frac{x ^{\frac{3}{2}} - 3 x + 12 x ^{2} - 8 x + 4 x x}{2}