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Popular >

derivative of ln(x^{1/x})

Solution

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

Solution

\frac{- ln ( x ) + 1}{x ^{2}}

Solution steps

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

Apply the chain rule:

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

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

\frac{d}{d x} (x^{\frac{1}{x}}) = x^{\frac{1 - 2 x}{x}} (- ln (x) + 1)

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

Simplify

\frac{1}{x ^{\frac{1}{x}}} x^{\frac{1 - 2 x}{x}} (- ln (x) + 1) : \frac{- ln ( x ) + 1}{x ^{2}}

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

Popular Examples

integral of (2/3 sec(x)tan(x))

\int (\frac{2}{3} sec (x) tan (x)) d x

integral of 5e^{4x}

\int 5 e^{4 x} d x

integral of (x+1)/(x-5)

\int \frac{x + 1}{x - 5} d x

integral of (2x)/(sqrt(2x^2+5y^2))

\int \frac{2 x}{2 x ^{2} + 5 y ^{2}} d x

integral of 1/((x+1)ln^2(x+1))

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

Frequently Asked Questions (FAQ)

What is the derivative of ln(x^{1/x}) ?
The derivative of ln(x^{1/x}) is (-ln(x)+1)/(x^2)
What is the first derivative of ln(x^{1/x}) ?
The first derivative of ln(x^{1/x}) is (-ln(x)+1)/(x^2)