$$f(x)=(x+(x+x^{\frac 12})^{\frac 12})^{\frac 12}$$
$$f'(x)=\frac{1}{2}.(x+(x+x^{\frac 12})^{\frac 12})^{\frac{-1}{2}}.(x+(x+x^{\frac 12})^{\frac 12})'$$
$$f'(x)=\frac{1}{2}.(x+(x+x^{\frac 12})^{\frac 12})^{\frac{-1}{2}}.(1+\frac{1}{2}(x+x^{\frac 12})^{\frac{-1}{2}})(x+x^{\frac 12})'$$
$$f'(x)=\frac{1}{2}.(x+(x+x^{\frac 12})^{\frac 12})^{\frac{-1}{2}}.(1+\frac{1}{2}(x+x^{\frac 12})^{\frac{-1}{2}})(1+\frac 12.x^{\frac{-1}{2}})$$