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

The answer to (\partial)/(\partial x)(\sqrt[3]{x^2y+3y}) is (2xy)/(3(x^2y+3y)^{2/3)}

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

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

Solution steps

Treat

y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

What is (\partial)/(\partial x)(\sqrt[3]{x^2y+3y}) ?
The answer to (\partial)/(\partial x)(\sqrt[3]{x^2y+3y}) is (2xy)/(3(x^2y+3y)^{2/3)}