derivative of x^{1/4}y^{3/4}

解

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

\frac{y + 3 x \frac{d y}{d x}}{4 x ^{\frac{3}{4}} y ^{\frac{1}{4}}}

解答ステップ

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

積の法則を適用:

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

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

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

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

\frac{d}{d x} (y^{\frac{3}{4}}) = \frac{3 \frac{d y}{d x}}{4 y ^{\frac{1}{4}}}

= \frac{1}{4 x ^{\frac{3}{4}}} y^{\frac{3}{4}} + \frac{3 \frac{d y}{d x}}{4 y ^{\frac{1}{4}}} x^{\frac{1}{4}}

簡素化

\frac{1}{4 x ^{\frac{3}{4}}} y^{\frac{3}{4}} + \frac{3 \frac{d y}{d x}}{4 y ^{\frac{1}{4}}} x^{\frac{1}{4}} : \frac{y + 3 x \frac{d y}{d x}}{4 x ^{\frac{3}{4}} y ^{\frac{1}{4}}}

= \frac{y + 3 x \frac{d y}{d x}}{4 x ^{\frac{3}{4}} y ^{\frac{1}{4}}}