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

The derivative of ln(x^{ln(x})) is (2e^{ln^2(x)}ln(x))/(x^{ln(x)+1)}

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

The first derivative of ln(x^{ln(x})) is (2e^{ln^2(x)}ln(x))/(x^{ln(x)+1)}

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

\frac{2 e ^{l n^{2} (x)} ln ( x )}{x ^{l n (x) + 1}}

Solution steps

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

Apply the chain rule:

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

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

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

= \frac{1}{x ^{l n (x)}} \cdot \frac{2 e ^{l n^{2} (x)} ln ( x )}{x}

Simplify

\frac{1}{x ^{l n (x)}} \cdot \frac{2 e ^{l n^{2} (x)} ln ( x )}{x} : \frac{2 e ^{l n^{2} (x)} ln ( x )}{x ^{l n (x) + 1}}

= \frac{2 e ^{l n^{2} (x)} ln ( x )}{x ^{l n (x) + 1}}

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