derivative of x^{1/3}y^{2/3}

解答

\frac{d}{d x} (x^{\frac{1}{3}} y^{\frac{2}{3}})

\frac{y + 2 x \frac{d y}{d x}}{3 x ^{\frac{2}{3}} y ^{\frac{1}{3}}}

求解步骤

\frac{d}{d x} (x^{\frac{1}{3}} y^{\frac{2}{3}})

使用微分乘法定则:

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

f = x^{\frac{1}{3}}, g = y^{\frac{2}{3}}

= \frac{d}{d x} (x^{\frac{1}{3}}) y^{\frac{2}{3}} + \frac{d}{d x} (y^{\frac{2}{3}}) x^{\frac{1}{3}}

\frac{d}{d x} (x^{\frac{1}{3}}) = \frac{1}{3 x ^{\frac{2}{3}}}

\frac{d}{d x} (y^{\frac{2}{3}}) = \frac{2 \frac{d y}{d x}}{3 y ^{\frac{1}{3}}}

= \frac{1}{3 x ^{\frac{2}{3}}} y^{\frac{2}{3}} + \frac{2 \frac{d y}{d x}}{3 y ^{\frac{1}{3}}} x^{\frac{1}{3}}

化简

\frac{1}{3 x ^{\frac{2}{3}}} y^{\frac{2}{3}} + \frac{2 \frac{d y}{d x}}{3 y ^{\frac{1}{3}}} x^{\frac{1}{3}} : \frac{y + 2 x \frac{d y}{d x}}{3 x ^{\frac{2}{3}} y ^{\frac{1}{3}}}

= \frac{y + 2 x \frac{d y}{d x}}{3 x ^{\frac{2}{3}} y ^{\frac{1}{3}}}

\frac{d}{d x} (\frac{1}{2} x^{2} - x + \frac{1}{6} x^{3})

\int x^{2} \cdot x d x

\int \frac{x}{( x + 1 ) ( x + 2 ) ( x + 3 ) ^{2}} d x

y^{^{'}} + 2 y = e^{t}

\int (3 - x) 7^{(3 - x)^{2}} d x