What is the derivative of x^{1/3}y^{2/3} ?

The derivative of x^{1/3}y^{2/3} is (y+2x\frac{dy)/(dx)}{3x^{2/3}y^{1/3}}

What is the first derivative of x^{1/3}y^{2/3} ?

The first derivative of x^{1/3}y^{2/3} is (y+2x\frac{dy)/(dx)}{3x^{2/3}y^{1/3}}

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\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}}}

Solution steps

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

Apply the Product Rule:

(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}}

Simplify

\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}

f (x) = - 6 x^{2} - 3 x - 5 x^{3}

What is the derivative of x^{1/3}y^{2/3} ?
The derivative of x^{1/3}y^{2/3} is (y+2x\frac{dy)/(dx)}{3x^{2/3}y^{1/3}}
What is the first derivative of x^{1/3}y^{2/3} ?
The first derivative of x^{1/3}y^{2/3} is (y+2x\frac{dy)/(dx)}{3x^{2/3}y^{1/3}}