How to differentiate $x^{\log(x)}$? I want to differentiate $x^{\log(x)}$ with respect to $x$.
By using chain rule ($x{log(x)}\frac{1}{x}$), I got the answer $\log x$. Is it right?

Hint: A common trick in these sorts of problems is to remember that $a = e^{\log a}$  (where here $\log$ denotes the natural logarithm)
Now, setting $a=x^{\log x}$, you have:
$x^{\log x} = e^{\log (x^{\log x})}$, which by properties of logarithms simplifies to $=e^{(\log x)(\log x)}$.
Now see if you can continue.

As per your comment above, if $\log$ is meant to be of a different base, say $\log_b$, the same tactic works.  You would have $e^{(\log_b x)(\ln x)}$ instead.
