Differentiation of $x^{\sqrt{x}}$, how?

Question

The answer is (I think) $x^{\sqrt{x}-0.5} (1+0.5\ln(x))$, but how?

Answer 1

The way to differentiate most expressions of the form $x^{f(x)}$ is to rewrite them as $e^{f(x)\ln(x)},$ and then use the chain and product rules, and that works in this case too.

Answer 2

Let's write $E'$ to mean the derivative of the expression $E$.

Then in general: $$\begin{align} u^v & = e^{v\log u} \\ (u^v)' &= \left(e^{v\log u}\right)'\\ &=e^{v\log u}\cdot (v\log u)'\\ &=u^v \left(v'\log u + v\frac {u'}u\right) \\ &=u^vv'\log u + u^{v-1}vu' \end{align}$$

To check, try taking one of $u$ or $v$ to be the identity function and the other to be a constant.