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

解

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

\frac{5 x ^{2} - 12 x + 4}{2 x}

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

\frac{1}{2 x} (2 - x)^{2} + (- 2 (2 - x)) x : \frac{5 x ^{2} - 12 x + 4}{2 x}

= \frac{5 x ^{2} - 12 x + 4}{2 x}