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

The answer to (\partial)/(\partial x)(\sqrt[3]{x^2}) is 2/(3x^{1/3)}

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

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

\frac{2}{3 x ^{\frac{1}{3}}}

Solution steps

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

Apply the chain rule:

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

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

\frac{\partial}{\partial x} (x^{2}) = 2 x

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

Simplify

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

= \frac{2}{3 x ^{\frac{1}{3}}}

\int \frac{( x ^{2} + 2 x + 1 )}{x} d x

\int \frac{e ^{2 x}}{9 + e ^{4 x}} d x

\int \frac{1}{x \cdot ( x ^{2} - 1 ) ^{\frac{1}{2}}} d x

\frac{d}{d x} (3 x^{3} - \frac{1}{x ^{3}})

e^{- 4 t} \cdot cos (3 t)

What is (\partial)/(\partial x)(\sqrt[3]{x^2}) ?
The answer to (\partial)/(\partial x)(\sqrt[3]{x^2}) is 2/(3x^{1/3)}