derivative of (x-1*sqrt(x^2-2x+2))

解

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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