Differentiate $x^{ln x}$ with respect to $ln x$ .

(x^{l n x}) (l n x)

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

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(x^{l n x}) (l n x) / x

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- ({l n x}^{l n x}) (l n x) / x

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Solution

The correct option is A 2 $(x^{l n x}) (l n x)$
Let $y = x^{ln x}$ and $z = ln x \Rightarrow x = e^{z}$ we have to find $\frac{d y}{d z}$
$\Rightarrow y = (e^{z})^{z} = e^{z^{2}}$
$\Rightarrow \frac{d y}{d z} = e^{z^{2}} . (2 z) = x^{ln x} . (2 ln x)$

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