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

解

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

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

解答ステップ

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

積の法則を適用:

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

f = (x - 1)^{\frac{1}{3}}, g = (x + 2)^{\frac{2}{3}}

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

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

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

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

簡素化

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

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