derivative of x^{0.5}*y^{0.5}

解

\frac{d}{d x} (x^{0.5} \cdot y^{0.5})

\frac{0.5 y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}} + \frac{0.5 x ^{\frac{1}{2}} \frac{d y}{d x}}{y ^{\frac{1}{2}}}

解答ステップ

\frac{d}{d x} (x^{0.5} y^{0.5})

積の法則を適用:

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

f = x^{0.5}, g = y^{0.5}

= \frac{d}{d x} (x^{0.5}) y^{0.5} + \frac{d}{d x} (y^{0.5}) x^{0.5}

\frac{d}{d x} (x^{0.5}) = \frac{0.5}{x ^{0.5}}

\frac{d}{d x} (y^{0.5}) = \frac{0.5 \frac{d y}{d x}}{y ^{\frac{1}{2}}}

= \frac{0.5}{x ^{0.5}} y^{0.5} + \frac{0.5 \frac{d y}{d x}}{y ^{\frac{1}{2}}} x^{0.5}

簡素化

\frac{0.5}{x ^{\frac{1}{2}}} y^{\frac{1}{2}} + \frac{0.5 \frac{d y}{d x}}{y ^{\frac{1}{2}}} x^{\frac{1}{2}} : \frac{0.5 y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}} + \frac{0.5 x ^{\frac{1}{2}} \frac{d y}{d x}}{y ^{\frac{1}{2}}}

= \frac{0.5 y ^{\frac{1}{2}}}{x ^{\frac{1}{2}}} + \frac{0.5 x ^{\frac{1}{2}} \frac{d y}{d x}}{y ^{\frac{1}{2}}}