derivative of x^{2/3}(5-x)

解

\frac{d}{d x} (x^{\frac{2}{3}} (5 - x))

\frac{10 - 5 x}{3 x ^{\frac{1}{3}}}

解答ステップ

\frac{d}{d x} (x^{\frac{2}{3}} (5 - x))

積の法則を適用:

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

f = x^{\frac{2}{3}}, g = 5 - x

= \frac{d}{d x} (x^{\frac{2}{3}}) (5 - x) + \frac{d}{d x} (5 - x) x^{\frac{2}{3}}

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

\frac{d}{d x} (5 - x) = - 1

= \frac{2}{3 x ^{\frac{1}{3}}} (5 - x) + (- 1) x^{\frac{2}{3}}

簡素化

\frac{2}{3 x ^{\frac{1}{3}}} (5 - x) + (- 1) x^{\frac{2}{3}} : \frac{10 - 5 x}{3 x ^{\frac{1}{3}}}

= \frac{10 - 5 x}{3 x ^{\frac{1}{3}}}