What is the derivative of x^{-ln(x}) ?

The derivative of x^{-ln(x}) is -(2e^{-ln^2(x)}ln(x))/x

What is the first derivative of x^{-ln(x}) ?

The first derivative of x^{-ln(x}) is -(2e^{-ln^2(x)}ln(x))/x

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\frac{d}{d x} (x^{- l n (x)})

- \frac{2 e ^{- l n^{2} (x)} ln ( x )}{x}

Solution steps

\frac{d}{d x} (x^{- l n (x)})

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{- l n (x)} = e^{(- l n (x)) l n (x)}

= \frac{d}{d x} (e^{(- l n (x)) l n (x)})

Apply the chain rule:

e^{(- l n (x)) l n (x)} \frac{d}{d x} ((- ln (x)) ln (x))

= e^{(- l n (x)) l n (x)} \frac{d}{d x} ((- ln (x)) ln (x))

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

= e^{(- l n (x)) l n (x)} (- \frac{2 ln ( x )}{x})

Simplify

e^{(- l n (x)) l n (x)} (- \frac{2 ln ( x )}{x}) : - \frac{2 e ^{- l n^{2} (x)} ln ( x )}{x}

= - \frac{2 e ^{- l n^{2} (x)} ln ( x )}{x}

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