Finding the derivative of $x$ to the power something that is a function of $x$ if $y = x^{(x+1)^\frac12}$
then how can I get the first derivative of $y$?

$$y=x^{(x+1)^{1/2}}=x^{\sqrt{x+1}}=e^{\sqrt{x+1}\log x}$$
and since
$$(\sqrt{x+1}\log x)'=\frac{\log x}{2\sqrt{x+1}}+\frac{\sqrt{x+1}}{x}=\frac{x\log x+2x+2}{2x\sqrt{x+1}}$$
we get, applying as suggested the chaing rule, that
$$y'=y\frac{x\log x+2x+2}{2x\sqrt{x+1}}=x^{\sqrt{x+1}}\cdot\frac{x\log x+2x+2}{2x\sqrt{x+1}}$$
