What is the derivative of ln(x^{1/5}) ?

The derivative of ln(x^{1/5}) is 1/(5x)

What is the first derivative of ln(x^{1/5}) ?

The first derivative of ln(x^{1/5}) is 1/(5x)

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\frac{d}{d x} (ln (x^{\frac{1}{5}}))

\frac{1}{5 x}

Solution steps

\frac{d}{d x} (ln (x^{\frac{1}{5}}))

Apply the chain rule:

\frac{1}{x ^{\frac{1}{5}}} \frac{d}{d x} (x^{\frac{1}{5}})

= \frac{1}{x ^{\frac{1}{5}}} \frac{d}{d x} (x^{\frac{1}{5}})

\frac{d}{d x} (x^{\frac{1}{5}}) = \frac{1}{5 x ^{\frac{4}{5}}}

= \frac{1}{x ^{\frac{1}{5}}} \cdot \frac{1}{5 x ^{\frac{4}{5}}}

Simplify

\frac{1}{x ^{\frac{1}{5}}} \cdot \frac{1}{5 x ^{\frac{4}{5}}} : \frac{1}{5 x}

= \frac{1}{5 x}

What is the derivative of ln(x^{1/5}) ?
The derivative of ln(x^{1/5}) is 1/(5x)
What is the first derivative of ln(x^{1/5}) ?
The first derivative of ln(x^{1/5}) is 1/(5x)