What is the derivative of ln(x^{1/3}) ?

The derivative of ln(x^{1/3}) is 1/(3x)

What is the first derivative of ln(x^{1/3}) ?

The first derivative of ln(x^{1/3}) is 1/(3x)

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

\frac{1}{3 x}

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

= \frac{1}{3 x}

What is the derivative of ln(x^{1/3}) ?
The derivative of ln(x^{1/3}) is 1/(3x)
What is the first derivative of ln(x^{1/3}) ?
The first derivative of ln(x^{1/3}) is 1/(3x)