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

解

導関数

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

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

解答ステップ

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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