What is (\partial)/(\partial x)(\sqrt[3]{1/x}) ?

The answer to (\partial)/(\partial x)(\sqrt[3]{1/x}) is -1/(3x^{4/3)}

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(\partial)/(\partial x)(\sqrt[3]{1/x})

- \frac{1}{3 x ^{\frac{4}{3}}}

Solution steps

Apply the chain rule:

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

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

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

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

Simplify

\frac{1}{3 ( \frac{1}{x} ) ^{\frac{2}{3}}} (- \frac{1}{x ^{2}}) : - \frac{1}{3 x ^{\frac{4}{3}}}

= - \frac{1}{3 x ^{\frac{4}{3}}}

x \to 1 lim (- cos (\frac{1}{x - 1}))

\frac{\partial}{\partial x} (6 e^{x} cos (yz))

\frac{d}{d x} (5 + sin (45))

x \to 0 + lim (\frac{x ^{2}}{5} - \frac{7}{x})

What is (\partial)/(\partial x)(\sqrt[3]{1/x}) ?
The answer to (\partial)/(\partial x)(\sqrt[3]{1/x}) is -1/(3x^{4/3)}