What is (\partial)/(\partial y)((x^3+(y/2))^{2/3}) ?

The answer to (\partial)/(\partial y)((x^3+(y/2))^{2/3}) is (2^{1/3})/(3(2x^3+y)^{1/3)}

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(\partial)/(\partial y)((x^3+(y/2))^{2/3})

\frac{\partial}{\partial y} ((x^{3} + (\frac{y}{2}))^{\frac{2}{3}})

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

Solution steps

\frac{\partial}{\partial y} ((x^{3} + (\frac{y}{2}))^{\frac{2}{3}})

Simplify

(x^{3} + (\frac{y}{2}))^{\frac{2}{3}} : (\frac{2 x ^{3} + y}{2})^{\frac{2}{3}}

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

Treat

x

as a constant

Apply the chain rule:

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

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

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

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

Simplify

\frac{2}{3 ( \frac{2 x ^{3} + y}{2} ) ^{\frac{1}{3}}} \cdot \frac{1}{2} : \frac{2 ^{\frac{1}{3}}}{3 ( 2 x ^{3} + y ) ^{\frac{1}{3}}}

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

\frac{d}{d x} (765480)

y^{^{''}} + y^{^{'}} - 3 y = 0

\int 10 x e x^{2} d x

\frac{d}{d x} (800 - 800 x^{- 1})

\frac{\partial}{\partial t} (\frac{sin ( t )}{t})

What is (\partial)/(\partial y)((x^3+(y/2))^{2/3}) ?
The answer to (\partial)/(\partial y)((x^3+(y/2))^{2/3}) is (2^{1/3})/(3(2x^3+y)^{1/3)}