無理関数√(x+√x)を微分する問題

$$\begin{aligned} \hspace{10pt}y' & =\frac{1}{2} (x+x^\frac{1}{2})^{-\frac{1}{2}}(x+x^\frac{1}{2})'\hspace{10pt}\\[1em] & = \frac{1}{2} (x+x^\frac{1}{2})^{-\frac{1}{2}}\left(1+\frac{1}{2}x^{-\frac{1}{2}}\right)\hspace{10pt}\\[1em] & = \frac{1}{2}\frac{1}{\sqrt{x+\sqrt{x}}}\left(1+\frac{1}{2\sqrt{x}}\right)\\[1em] & = \frac{1}{2}\frac{1}{\sqrt{x+\sqrt{x}}}\frac{2\sqrt{x}+1}{2\sqrt{x}}\\[1em] & =\frac{2\sqrt{x}+1}{4\sqrt{x^2+x\sqrt{x}}}\\[1em] \end{aligned}$$