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

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

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

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

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

Solution steps

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

Treat

x

as a constant

Apply the chain rule:

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

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

\frac{\partial}{\partial y} (x + 2 y^{3}) = 6 y^{2}

= \frac{1}{3 ( x + 2 y ^{3} ) ^{\frac{2}{3}}} \cdot 6 y^{2}

Simplify

\frac{1}{3 ( x + 2 y ^{3} ) ^{\frac{2}{3}}} \cdot 6 y^{2} : \frac{2 y ^{2}}{( x + 2 y ^{3} ) ^{\frac{2}{3}}}

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

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