derivative of x^{0.4}y^{0.6}

解

\frac{d}{d x} (x^{0.4} y^{0.6})

\frac{0.4 y ^{0.6}}{x ^{0.6}} + \frac{0.6 x ^{0.4} \frac{d y}{d x}}{y ^{0.4}}

解答ステップ

\frac{d}{d x} (x^{0.4} y^{0.6})

積の法則を適用:

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

f = x^{0.4}, g = y^{0.6}

= \frac{d}{d x} (x^{0.4}) y^{0.6} + \frac{d}{d x} (y^{0.6}) x^{0.4}

\frac{d}{d x} (x^{0.4}) = \frac{0.4}{x ^{0.6}}

\frac{d}{d x} (y^{0.6}) = \frac{0.6 \frac{d y}{d x}}{y ^{0.4}}

= \frac{0.4}{x ^{0.6}} y^{0.6} + \frac{0.6 \frac{d y}{d x}}{y ^{0.4}} x^{0.4}

簡素化

\frac{0.4}{x ^{0.6}} y^{0.6} + \frac{0.6 \frac{d y}{d x}}{y ^{0.4}} x^{0.4} : \frac{0.4 y ^{0.6}}{x ^{0.6}} + \frac{0.6 x ^{0.4} \frac{d y}{d x}}{y ^{0.4}}

= \frac{0.4 y ^{0.6}}{x ^{0.6}} + \frac{0.6 x ^{0.4} \frac{d y}{d x}}{y ^{0.4}}