Math Help - seemingly simple derivative

1. seemingly simple derivative

hello all. my friend has asked me for help on a derivation but i've come up blank.

the derivation in question is:

$\frac{d}{dx}lnx^{lnx}$

i thought $lnx$

he thought $2\frac{lnx}{x}$

what do YOU think?

2. Originally Posted by makieru
hello all. my friend has asked me for help on a derivation but i've come up blank.

the derivation in question is:

$\frac{d}{dx}lnx^{lnx}$

i thought $lnx$

he thought $2\frac{lnx}{x}$

what do YOU think?
It's $\frac{2\ln x}{x}$. Here's why:

$\ln\left(x^{\ln x}\right)=(\ln x)(\ln x)=(\ln x)^2$

Use the chain rule:

$2\ln x\cdot\frac{1}{x}=\frac{2\ln x}{x}$