derivative of \sqrt[3]{x}(8-x)

解

\frac{d}{d x} (3 x (8 - x))

\frac{8 - x}{3 x ^{\frac{2}{3}}} - 3 x

解答ステップ

\frac{d}{d x} (3 x (8 - x))

積の法則を適用:

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

f = 3 x, g = 8 - x

= \frac{d}{d x} (3 x) (8 - x) + \frac{d}{d x} (8 - x) 3 x

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

\frac{d}{d x} (8 - x) = - 1

= \frac{1}{3 x ^{\frac{2}{3}}} (8 - x) + (- 1) 3 x

簡素化

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

= \frac{8 - x}{3 x ^{\frac{2}{3}}} - 3 x