[tex]f(x)=\sqrt{x+\sqrt{x+\sqrt x}}\\
f(x)=\sqrt{x+\sqrt{x+x^{\frac{1}{2}}}}\\
f(x)=\sqrt{x+\sqrt{x^{\frac{3}{2}}}}\\
f(x)=\sqrt{x+\left(x^{\frac{3}{2}}\right)^{\dfrac{1}{2}}}}\\
f(x)=\sqrt{x+x^{\frac{3}{4}}}}\\
f(x)=\left(x^{\frac{7}{4}}}\right)^{\frac{1}{2}}\\
f(x)=x^{\frac{7}{8}}}\\\\
f'(x)=\dfrac{7}{8}x^{-\frac{1}{8}}
[/tex]
