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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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