derivative of x^{0.7}y^{0.3}

解答

\frac{d}{d x} (x^{0.7} y^{0.3})

\frac{0.7 y ^{0.3}}{x ^{0.3}} + \frac{0.3 x ^{0.7} \frac{d y}{d x}}{y ^{0.7}}

求解步骤

\frac{d}{d x} (x^{0.7} y^{0.3})

使用微分乘法定则:

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

f = x^{0.7}, g = y^{0.3}

= \frac{d}{d x} (x^{0.7}) y^{0.3} + \frac{d}{d x} (y^{0.3}) x^{0.7}

\frac{d}{d x} (x^{0.7}) = \frac{0.7}{x ^{0.3}}

\frac{d}{d x} (y^{0.3}) = \frac{0.3 \frac{d y}{d x}}{y ^{0.7}}

= \frac{0.7}{x ^{0.3}} y^{0.3} + \frac{0.3 \frac{d y}{d x}}{y ^{0.7}} x^{0.7}

化简

\frac{0.7}{x ^{0.3}} y^{0.3} + \frac{0.3 \frac{d y}{d x}}{y ^{0.7}} x^{0.7} : \frac{0.7 y ^{0.3}}{x ^{0.3}} + \frac{0.3 x ^{0.7} \frac{d y}{d x}}{y ^{0.7}}

= \frac{0.7 y ^{0.3}}{x ^{0.3}} + \frac{0.3 x ^{0.7} \frac{d y}{d x}}{y ^{0.7}}

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