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Let $f(x)=x^x$.

What is the derivative of $f$?

This function can't be treated by chain rule or product rule or (e^x)'=e^x

  • 1
    hint: $x^x = e^{\log(x^x)} = e^{x\log(x)}$2011-10-10

3 Answers 3

14

$\begin{align*}\frac{d}{dx}(x^x) &= \frac{d}{dx}(e^{x \ln x}) & \textrm{(Using the fact that $x^x = e^{x \ln x})$}\\ & = e^{x \ln x} \frac{d}{dx}(x \ln x) & \textrm{(Using the chain rule)} \\ &=x^x (\ln x + 1) & \textrm{(Using the product rule)}\end{align*}$

  • 3
    One can in fact use this technique to derive the more general formula for $f(x)^{g(x)}$.2011-10-10
5

let $y=x^x$ $\implies$ $\log y=x\log x$ Differentiate on both sides $\dfrac{dy}{dx}\left(\dfrac{1}{y}\right)=\log x+1$ $\implies $ $\dfrac{dy}{dx}=x^x(\log x+1)$

3

$f(x)=x^x=e^{x\ln x}$ from here you use the chain rule for $(e^g)'=e^g g'$.