derivative of x^{1/3}(x^2-4)

解

\frac{d}{d x} (x^{\frac{1}{3}} (x^{2} - 4))

\frac{7 x ^{2} - 4}{3 x ^{\frac{2}{3}}}

解答ステップ

\frac{d}{d x} (x^{\frac{1}{3}} (x^{2} - 4))

積の法則を適用:

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

f = x^{\frac{1}{3}}, g = x^{2} - 4

= \frac{d}{d x} (x^{\frac{1}{3}}) (x^{2} - 4) + \frac{d}{d x} (x^{2} - 4) x^{\frac{1}{3}}

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

\frac{d}{d x} (x^{2} - 4) = 2 x

= \frac{1}{3 x ^{\frac{2}{3}}} (x^{2} - 4) + 2 x x^{\frac{1}{3}}

簡素化

\frac{1}{3 x ^{\frac{2}{3}}} (x^{2} - 4) + 2 x x^{\frac{1}{3}} : \frac{7 x ^{2} - 4}{3 x ^{\frac{2}{3}}}

= \frac{7 x ^{2} - 4}{3 x ^{\frac{2}{3}}}