$\dfrac{\mathrm d}{\mathrm dx}x^x=$?

Question

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

What is the derivative of $f$?

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

What is your definition of $x^x$? – Chris Eagle Oct 10 '11 at 18:17 — Chris Eagle, Oct 10 '11 at 18:17
hint: $x^x = e^{\log(x^x)} = e^{x\log(x)}$ – Andy Oct 10 '11 at 18:20 — Andy, Oct 10 '11 at 18:20

score 14 · Accepted Answer · answered Oct 10 '11 at 18:23

14

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

answered Oct 10 '11 at 18:23

sxd

3,504

3

One can in fact use this technique to derive the more general formula for $f(x)^{g(x)}$. – J. M. ain't a mathematician Oct 10 '11 at 22:49

score 5 · Answer 2 · answered Oct 10 '11 at 18:25

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 $ $\dfrac{dy}{dx}=x^x(\log x+1)$

answered Oct 10 '11 at 18:25

Ramana Venkata

2,076

Beni Bogosel · Answer 3 · 2013-03-10T17:39:12.957

3

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

edited Mar 10 '13 at 17:39

answered Oct 10 '11 at 18:21

Beni Bogosel

23,381

$\dfrac{\mathrm d}{\mathrm dx}x^x=$?

3 Answers3