Derivative of ln of x to the power ln x

Question

Differentiate $y=\ln\!\big(x^{\ln x}\big)$ with respect to $x$ (for $x>0$ ).

Short answer

$y'=\dfrac{2\ln x}{x}$ . Simplify first: $\ln(x^{\ln x})=(\ln x)^2$ , and the chain rule gives $2\ln x\cdot\tfrac1x$ .

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 3.1 Limits and Derivatives, 'Basics of derivatives', p.33; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

The Derivative as a Limit: Rates of Change and the Derivative Function
The Chain Rule