Implicit derivatives and logarithmic derivatives: $\left(x^{\sqrt{x}}\right)'=?$ How would I find the derivative with respect to $x$ of $$y = \left(x^{\sqrt{x}}\right)'.$$ I can find the correct answer using the method of logarithmic differentiation that my book  mysteriously suggest to use for some reason, BUT how can I find the derivative without using that method?
Furthermore how would I know when I should use logarithmic differentiation? 
And why can I not get the correct answer using the Chain rule only? 

You may write, for $x>0$,
$$
x^{\sqrt{x}}=e^{\sqrt{x}\ln x}
$$ then, you get
$$
\left(x^{\sqrt{x}}\right)'=(\sqrt{x}\ln x)'e^{\sqrt{x}\ln x}
$$ using
$$
(e^u)'=u'e^u.
$$ Finally, we have

$$
\left(x^{\sqrt{x}}\right)'=\left(1+\frac{\ln x}2\right)x^{\sqrt{x}-\frac12},\qquad x>0.
$$

