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

解

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

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

解答ステップ

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

積の法則を適用:

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

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

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

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

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

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

簡素化

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

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