derivative of x^{1/3}y^{2/3}

解

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

\frac{y + 2 x \frac{d y}{d x}}{3 x ^{\frac{2}{3}} y ^{\frac{1}{3}}}

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

\frac{1}{3 x ^{\frac{2}{3}}} y^{\frac{2}{3}} + \frac{2 \frac{d y}{d x}}{3 y ^{\frac{1}{3}}} x^{\frac{1}{3}} : \frac{y + 2 x \frac{d y}{d x}}{3 x ^{\frac{2}{3}} y ^{\frac{1}{3}}}

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