derivative \sqrt[5]{x}(x-2)

解

導関数

5 x (x - 2)

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

解答ステップ

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

積の法則を適用:

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

f = 5 x, g = x - 2

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

\frac{d}{d x} (5 x) = \frac{1}{5 x ^{\frac{4}{5}}}

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

= \frac{1}{5 x ^{\frac{4}{5}}} (x - 2) + 1 \cdot 5 x

簡素化

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

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