Answer
$$y'=\frac{1}{2\sqrt{x+\sqrt{x+\sqrt x}}}\left(1+\frac{1}{2\sqrt{x+\sqrt x}}\left(1+\frac{1}{2\sqrt x}\right)\right)$$
Work Step by Step
$$y'=\left(\sqrt{x+\sqrt{x+\sqrt x}}\right)'=\frac{1}{2\sqrt{x+\sqrt{x+\sqrt x}}}\left(x+\sqrt{x+\sqrt x}\right)'=\frac{1}{2\sqrt{x+\sqrt{x+\sqrt x}}}\left(1+\frac{1}{2\sqrt{x+\sqrt x}}(x+\sqrt x)'\right)=\frac{1}{2\sqrt{x+\sqrt{x+\sqrt x}}}\left(1+\frac{1}{2\sqrt{x+\sqrt x}}\left(1+\frac{1}{2\sqrt x}\right)\right)$$
