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

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

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

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

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derivative

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