$y=\sqrt{x+\sqrt{x+\sqrt{x}}}\\\to y'=\Bigg(\sqrt{x+\sqrt{x+\sqrt{x}}}\Bigg)'\\=\dfrac{1}{2\sqrt{x+\sqrt{x+\sqrt{x}}}}.(x+\sqrt{x+\sqrt{x}})'\\=\dfrac{1}{2\sqrt{x+\sqrt{x+\sqrt{x}}}}.\Bigg[1+\dfrac{1}{2\sqrt{x+\sqrt{x}}}.(x+\sqrt{x})'\Bigg]\\=\dfrac{1}{2\sqrt{x+\sqrt{x+\sqrt{x}}}}.\Bigg[1+\dfrac{1}{2\sqrt{x+\sqrt{x}}}.\Bigg(1+\dfrac{1}{2\sqrt{x}}\Bigg)\Bigg]\\=\dfrac{1}{2\sqrt{x+\sqrt{x+\sqrt{x}}}}.\Bigg(1+\dfrac{1}{2\sqrt{x+\sqrt{x}}}+\dfrac{1}{4\sqrt{x}{\sqrt{x+\sqrt{x}}}}\Bigg)\\=\dfrac{1}{2\sqrt{x+\sqrt{x+\sqrt{x}}}}.\Bigg(1+\dfrac{1}{2\sqrt{x+\sqrt{x}}}+\dfrac{1}{4{\sqrt{x^2+x\sqrt{x}}}}\Bigg)$